It's only "on the" in some special cases, namely dates. On monday = am Monday, on the first of july = am ersten Juli, on the third day = am dritten Tag. In all other cases, on the = auf dem; am = at.
how would you say "the woman drinks on the table" then?
die Frau trinkt auf dem Tisch? (just guessing here, I have no idea)
If the woman drank too much, probably she would drink on the table. :P It is "Die Frau trinkt auf dem Tisch", correct.
because part of the reason is on the table sounds like she is on the table
yes! am = an dem = at the. "Auf dem" cannot be shortened and it means "on the"
And how could one know if smth can or can't be shortened? Some rule or just remebering?
That's why I was confused. Isn't Tisch dative here? So wouldn't it be dem Tisch and so the sentence would be 'Die Frau trinkt an dem Tisch' ?
It is possible, but the meaning is different. It would mean the woman is drinking over by the table, rather than at the table.