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See also this question about sorcerers triggering wild magic surges when casting putatively sorcerer spells from magic items).

In conclusion, I would say that
**Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such.

If a feature allows you to know a spell and explicitly references a class, the spell can also be of that class, if you are of that class and you choose to cast it as such**.

See also this question about sorcerers triggering wild magic surges when casting putatively sorcerer spells from magic items.

In conclusion, I would say that
Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such.

If a feature allows you to know a spell and explicitly references a class, the spell can also be of that class, if you are of that class and you choose to cast it as such.

There are several ways to resolve this. There is a clear, direct answer: Any spell you cast that is on your class spell list counts as a spell of that class. This has the advantages of being simple, found in multiple places supporting each other, and a response directed at the question itself. It has the disadvantage of being largely based on (now unofficial) tweets from Jeremy Crawford. I choose not to follow that path for reasons other than it being unofficial, per se. First, this is a complex question and I don't think all the nuances and caveats lend themselves to explanation within a tweet character limit. Second, I am uncomfortable with a logical conclusion that follows - an individual spell you cast that is on two different spell lists of yours is then simultaneously a spell of both of those classes. And third, applied broadly it contradicts what is an official Sage Advice answer (see below). I think unofficial answers (tweets) have a valuable role in clarifying things when official answers aren't forthcoming, but I hesitate to accept them when they contradict what is official.

Another way to resolve this is to look at what is actually within the the text of official sources such as the PHB and Sage Advice. Unfortunately, this approach relies on interpreting passages that are not explicitly directed at the question itself ('what makes a spell of a certain class?'), and interpretations can vary. In particular, the passage that I find most helpful is in the rules on Multiclassing, which are themselves optional. I certainly understand the arguments of those who would apply it narrowly ('this section about spells is only for multiclassing and only if you agree to that option'), but I choose to interpret them broadly ('this optional section illustrates a larger point about how spells work that is true across the core system'). My choice is undeniably biased toward utility - I find a speculative principle that I can actually use better than stating that there is no clear answer to the larger question. While true, the latter is not particularly helpful, and now, six years after publication of the PHB, after numerous errata and Sage Advice clarifications, and after two large presentations of new rules (XGtE and TCoE) at least one of which deliberately clarified questions about timing, sleeping, AoE on a grid, etc., it seems likely that WoTC either considers this issue settled or has no interest in addressing it.

Here is the passage in full from the (optional) Multiclassing Rules

Objection: This section is optional, and specifically says it is to be used only "once you have the Spellcasting feature from more than one class".
Response: Yes, but it also distinguishes between rules (like those about spells known and prepared) that are true "as if you were a single-classed member of that class" and rules (like those about total spell slots) that apply only when you have multiple Spellcasting classes. I don't think it is a bridge too far to believe that the principles it clarifies 'as if you were single classed' also apply to when you actually are single-classed. As for it being optional, many of the times the greater question (how do you know when a spell is a class spell) comes up is in attempting to adjudicate optional features, like Feats. I will take guidance where I can find it.

In particular, in what follows I am assuming that by "each spell you know and prepare is associated with one of your classes" means that  If a feature that allows you to know a spell can be tied to a specific class, that spell counts as being of that class. Indeed, what else could it mean that would not be redundant or a tautology? What other kind of association is worth mentioning?

Sorcerer spells (for example) are spells off the sorcerer spell list that you know because you are a sorcerer (having the Spellcasting ability by virtue of you being in the sorcerer class).

"Sorcerer: Spellcasting" (PHB 101; emphasis mine)

Some racial feats tell you the class of the spells they are granting. "Wood Elf Magic" in XGtE

Others do not. "Drow High Magic" in XGtE

Magic items also typically say "you cast", "you can cast", etc. However, they do not usually say that you 'learn' or 'know' a spell, which is in clear contrast to the language used with feats.

Based on this, I would say that if the magic item allows you to cast a spell that you do not know, it is not a class spell. However, if the spell it is allowing you to cast is one that you already know, then you are casting a spell that you know from a specific class, and thus it is a spell of that class.

If you are multiclassed and have access to a spell from another class that is also on the sorcerer spell list, you still can't cast it as a sorcerer unless you know it through being a sorcerer ("Each spell you know and prepare is associated with one of your classes").

I realize that 'your spellcasting ability' (ie, Charisma) is not 'your ability to use Spellcasting' (ie, a class feature). But the precedent of, when a spell could be sourced from multiple classes, choosing the spellcasting ability at the time of casting seems a clear enough proxy for the class of the spell to me.

In conclusion, I would say that Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such.
If a feature allows you to know a spell and explicitly references a class, the spell can also be of that class, if you are of that class and you choose to cast it as such.

There are several possible ways to resolve this. There is a clear, direct answer: Any spell you cast that is on your class spell list counts as a spell of that class. This has the advantages of being simple, being found in multiple places all supporting each other, and being a response directed at the question itself. It has the disadvantage of being largely based on (now unofficial) tweets from Jeremy Crawford. I choose not to follow that path for reasons other than it being unofficial, per se. First, this is a complex question and I don't think all the nuances and caveats lend themselves to explanation within a tweet character limit. Second, I am uncomfortable with one logical conclusion that follows - an individual spell you cast that is on two different spell lists of yours is then simultaneously a spell of both of those classes. And third, it contradicts what is an official Sage Advice ruling (see below). I think unofficial answers (tweets) have a valuable role in clarifying things when official answers aren't forthcoming, but I hesitate to accept them when they contradict what is official.

Another way to resolve this is to look at what is actually within the the text of official sources such as the PHB and Sage Advice. Unfortunately, this approach relies on interpreting passages that are not explicitly directed at the question itself ('what makes a spell of a certain class?'), and interpretations can vary.

In particular, the passage that I find most helpful is in the rules on Multiclassing, which are themselves optional. I understand the arguments of those who would apply this passage narrowly ('this section about spells is only for PCs who multiclass, and only if you agree to that option'), but I choose to interpret it broadly ('this optional section illustrates a larger point about how spells work that is true across the core system'). My choice is undeniably biased toward utility - I find a speculative principle that I can actually use better than the interpretation that there is no clear answer to the larger question. The latter might be true, but it is not particularly helpful, and now, six years after publication of the PHB, after numerous errata and Sage Advice clarifications, and after two large presentations of new rules (XGtE and TCoE), at least one of which deliberately clarified questions about timing, sleeping, AoE on a grid, etc., it seems likely that WoTC either considers this issue settled or has no interest in addressing it.

Here is the passage in full from the Multiclassing Rules:

Objection: This section is optional, and specifically says it is to be used only "once you have the Spellcasting feature from more than one class".
Response: Yes, but it also distinguishes between rules (like those about spells known and prepared) that are true "as if you were a single-classed member of that class" and rules (like those about total spell slots) that apply only when you have multiple Spellcasting classes. I don't think it is unreasonable to believe that the principles it clarifies 'as if you were single classed' also apply to when you actually are single-classed. As for it being optional, many of the times the greater question of 'how do you know when a spell is a class spell' actually comes up is when attempting to adjudicate other optional features, like Feats. I will take guidance where I can find it.

In what follows I am assuming that by "each spell you know and prepare is associated with one of your classes" means that:
If a feature that allows you to know a spell can be tied to a specific class, that spell counts as being of that class.
Indeed, what else could it mean that would not be redundant or a tautology? What other kind of association is worth mentioning?

Class spells are spells off the class spell list that you know because you are that class (you have the Spellcasting ability by virtue of you being in that class).

For example; "Sorcerer: Spellcasting" (PHB 101; emphasis mine)

Some racial feats tell you the class of the spells they are granting. 
"Wood Elf Magic" in XGtE

Others do not. 
"Drow High Magic" in XGtE

Magic items also typically say "you cast", "you can cast", etc in their descriptions. However, they do not usually say that you 'learn' or 'know' a spell, which is in clear contrast to the language used with feats.

Based on this, I would say that if the magic item allows you to cast a spell that you do not know, it is not a class spell. However, if the spell it is allowing you to cast is one that you already know, then you are casting a spell that you know from a specific class, and thus it is a spell of that class.

If you are multiclassed and have access to a spell from another class that is also on the sorcerer spell list, for example, you still can't cast it as a sorcerer unless you know it through being a sorcerer ("Each spell you know and prepare is associated with one of your classes").

I realize that 'your spellcasting ability' (ie, Charisma) is not 'your ability to use Spellcasting' (ie, a class feature). But the precedent of, when a spell could be sourced from multiple classes, choosing the spellcasting ability at the time of casting seems a clear enough proxy for the class of the spell to me.

In conclusion, I would say that
**Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such.

If a feature allows you to know a spell and explicitly references a class, the spell can also be of that class, if you are of that class and you choose to cast it as such**.

Some clear, some not

In conclusion, I would say that Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such. If a feature gives you the ability to cast a spell and explicitly references the class, the spell can also be of that class, if you choose to cast it as such.

Some clear cases, some not

In conclusion, I would say that Any spell on a class spell list that you know from being that class is a class spell, if you choose to cast it as such.
If a feature allows you to know a spell and explicitly references a class, the spell can also be of that class, if you are of that class and you choose to cast it as such.