The complicated stuff here is in english, not in spanish.
As you say, for native english people, 'teacup' or 'cup of tea' are used depending of the context.
The thing here in spanish is very easy:
¿Me das un vaso de agua?
We don't say:
¿Me das un agua vaso? or ¿Me das un vaso agua?
This biodegradable water bottle could replace plastic
The particle de is the one we use for spanish, so there's nothing to be worried about if you say (1). Number (2) is incorrect.
Also, often spanish speakers say
¿Me das un vaso con agua?
which is also correct, but saying 'vaso de agua' is spanish is like a glass made of water, which is absurd and nobody's gonna misunderstand that.
Saying 'vaso de agua' or 'vaso con agua' are often used and there's no problem with others like 'botella de agua' or 'botella con agua,' but remember it can't be 'botella agua' or 'agua botella.'