Have tried emailing Mark Levine?

Tonghang Zhou wrote his Purdue thesis on this topic in the early 1990's. docs.lib.purdue.edu/dissertations/AAI9301421 It's unpublished but you should be able to get a copy but the usual means.

I believe that Zariski studied with Castelnuovo, but eventually became dissatisfied with the level of rigour used in the Italian school. so he might be an example of what you are looking for in 2.

If $K=\mathbb{C}$ (and you reduce to this case) and $S$ is connected, the fibres $X_s$ and $X_{s'}$ are diffeomorphic by Ehresmann's theorem. So the de Rham cohomologies are isomorphic. If you choose a path from $s$ to $s'$, parallel translation along it wrt Gauss-Manin will give a specific isomorphism. Perhaps this is what you had in mind. Try looking at Voisin's Hodge theory books for details.

I have no idea about the history (which is I realize is your question), but speaking for myself, if I have to recall the definition quickly to a student, I'm going to pick a direction at random. However, if I had time to think about it, I would probably decide that since coordinates are (local) functions on the manifold, the "modern" approach makes more sense.

I'm assuming the second author is Noriko Yui. You could just email her.

@KingofPomelo OK, I see. I'll think about it, when I get a chance, but it isn't obvious.