I've seen that there is several kinds of PRF, and sometimes people speak about strong PRFs.
When proofs of a protocol or algorithm are given based on the assumption of the use of a PRF, are they speak implicitly about strong PRFs?
I disagree with the @RickyDemer. The concept of a strong PRF has specific technical meaning and is distinct from a PRF. This can be verified, say, in the Katz-Lindell textbook or by searching for the terms and looking at the lecture notes and papers that turn up. An important case where this distinction comes up is with 3-round versus 4-round Feistel networks. It is also a common exercise to build a PRF that is not a strong PRF.
There is a little variation on usage, as some theoretical crypto papers will only say "PRF" when they mean "strong PRF", but the vast majority distinguish between the terms.