The meaning you suggested is slightly different. If you translate this as "water bottle" it refers to a meaning such "the bottle to be filled 'only with' water", which actually would translate to Italian as "una bottiglia DA acqua/d'acqua". And yet, if you translate this sentence as "a water of bottle" it refers to any bottle in which you can put either water, or juice and so on. It is a very slight difference but important. So maybe that is why your answer "water bottle" is not accepted. So Italian is also quite meticulous about intricate meanings as English is.
If you use "dell'acqua" within the sentence it must precede a 'determinative' noun. It is like, you refer to the certain resource of water, or more simply, you talk about a certain brand of water. It can also refer to the category of uncountable nouns, which is another subject. In other words, it may depend on the context. You just accept that when a noun precedes another noun with undeterminative noun, it should be combined with an undeterminative as well.
I think a water bottle is just a plastic product meant to contain water (but may contain any liquid you want). This plastic product thing may be full of water but it may be empty (why not? someone may have drunk the water from it already).
If you mean a water that contains water, you need to say a bottle of water. This is what I guess, but please note I'm not a native English speaker so I could be wrong.
If you marched up to someone you probably wouldn't just demand "a bottle of water please", you would say "could I have..." "may I have..." etc. But if somebody asks "what would you like?" then you can simply reply "a bottle of water please" and it's not rude. In fact the alternative would sound a bit overly-formal in that case.