These words use the normal English meaning
D&D is written in English, and unless a term is defined in the rules as a game term, words in the rules use their normal, English meaning. Holding, carrying, and wearing are not explicitly given a special definition in the rules, so they use their normal, English meaning.
Dictionaries often cite multiple definitions with different meaning. Which one applies then needs to be seen in the context of how it is used in the rules (and in some cases, also interpreted more generously, because dictionary writers normally do not consider the possibility of monsters and magic that are real in the game).
Carrying means you have them the item anywhere on your person, either holding it in your hands, having it stowed in your backpack, wearing it etc. In the game rules, creatures have a Carrying Capacity (pPHB, p. 176) that limits how much they can carry
Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry
So anything that adds to the weight you carry is something you carry (even if the amount of weight added is neglected because it is too low). See also What items are considered "worn or carried"?, which has a top answer that concludes that anything on your characters sheet that you haven't dropped counts as worn or carried.
(Several rules refer to things you are "holding or carrying" or that are "worn or carried", which you might think suggests that something that you hold or wear is not being carried, as carrying would already include holding or wearing, so why even mention them? However, the rules are not a treatise on logic, and the "or" in natural language is not generally an exclusive either/or, it also can be an and/or.)
Wearing. While the dictionary quotes things like "carry on your person", I think this demands a bit more - while you carry a magical ring or a chainmail in your backpack, it will not work its magic or protect you. Magic items explicitly state that you have to put them on (DMG, DMG p. 140):
Using a magic item's properties might mean wearing or wielding it. A magic item meant to be worn must be donned in the intended fashion: boots go on the feet, gloves on the hands, hats and helmets on the head, and rings on the finger. Magic armor must be donned, a shield strapped to the arm, a cloak fastened about the shoulders. A weapon must be held.
Holding. In the dictionary this also encompasses the wider sense of owning something, but in the game it usually refers to holding something in your hand. In some cases, this is made explicit (e.g. the two-weapon fighting rules say they work with a melee weapon that you're holding in one hand, and a different one you're holding in the other hand), in other cases it is implicit , e.g. in the command spell or the Unconscious condition, which say "the target drops whatever it is holding".
To your specific question: while you are wearing the vial, you are already carrying it, and don't need to spend your free object interaction to carry it.
