Does "на озері" mean "on the lake" in the sense of being out on the lake in a boat, or does it also mean it in the sense of being on the shore of the lake, or by the lake - which we would usually say as "at the lake"? If not, what would be the translation gor "at the lake"?
Yes, it means both "on the lake" and "at the lake" depending on the context.
The friend could be out waterskiing ;)
I'm guessing the answer to Kags's question is that it means either on the lake like in a boat or at the lake (as opposed to at home or anywhere else). However, it would be great to get an authoritative explanation.