"Your green cup is empty."
Translation:A te zöld csészéd üres.
Is there a difference in the usage of "a csészéd" and "a te csészéd"? Are they always interchangeable or can they only be used in certain contexts?
in certain situations "te" can be obligatory, such as in "A te csészéd teljesen üres, az enyém csak félig." but it's mostly interchangeable. In particular, "te" can be added anytime.
The only things that come to my mind is when you contrast between possessors (like in my example above) & when you really want to stress that who the possessor is.
Yes, basically, when you emphasize the owner. Focus on "te", therefore you do not omit it. Whatever you emphasize in any sentence can't be omitted. That is only natural.
Omitted, because the emphasis is on the objects:
This is my cup, not my glass. - Ez a csészém, nem a poharam.
Not omitted, because the emphasis is on the possessor:
This is my car, not yours. - Ez az én autóm, nem a tiéd.