Translation:Do not keep your keys in the bag.
I'm a native speaker of Portuguese, where the exact same phenomenon occurs. In general, without context given, it is hard to tell. It is generally assumed that the bag does belong to the listener, but it does not have to be that way. It is all context-dependent, really. I can imagine, though, a situation in which the listener might assume the speaker is talking about the listener's bag, but the speaker is indeed talking about another bag. That, of course, can be fixed by just clarifying which bag. :) Hope this helped.
Yes, the short version would be answered the same way: «Non tenere le tue chiavi nella borsa.». The long version would be less ambiguous: «Un vizio/cattivo abitudine è non tenere le tue chiavi nella borsa.». The «è» being the main verb in the sentence clarifies the semantic role of «tenere».