First, you should know that partitive isn't really a part of Russian grammar anymore and it is not even taught in Russian schools any longer, instead it's generally classified as genetive. Yes, you can say:

Налей чаю or налей чая, instead of just налей чай and more and more people tend to prefer the latter nowadays. Therefore there is hardly any difference between those. A very slight nuance is there though. You see, mass noun can't be counted, therefore such nouns can be percieved like a single entity and partitive was invented to say that you only need a part of this entity:

Налей чай - чай can be viewed as a single entity, thus all the tea in the world there is
Налей чаю - just a part of this entity, not the whole amount of it

This simple way of thinking persist across all cases of partitive use:

Намешай бетону, насыпь сахару, отрежь сыру.

Yours - чашка чая, чашка чаю. I bet you can tell the difference yourself now. Just remember that there is hardly any really and you can choose whatever version you like more, although the latter is a bit more Russian, if you ask me


-What about the times where there isn't a determining noun to mark the phrase? How can one tell how the чай fits grammatically? Must we just assume BY CONTEXT that выпить will probably have a liquid based object?

Выпить чай - pure genetive, the others are genetive partitive, but again all are genetive. Выпить does have to have a liquid based object to make any sense, though it can be an indirect object --- Выпить чашку чаю. If you said something like, выпить шкаф that would first confuse me, then I'd assume that шкаф was liquified somehow

-Is modern partitive simply represented by genitive endings, even though the same trick is at work? ( Выпить чая )

Yes, выпить чаю, выпить чая are essentially the same with the same trick. Just remember that there can be a pure genetive instead: чашка чая for example vs. чашка чаю.

Is the partitive an noun-attributive or a verb-agreeing case? Meaning, for instance, Genitive is basically used to affect ~a noun~ with a noun, it's unrelated to the verb layout. Whereas Accusative is directly interworking with the sentence's verb. (a) and (b) show it being both types.

This is very hard to say. My best guess is that it's both. Выпить чаю - verb related, чашка чаю - noun + noun.