For as long as I've been monitoring this site (which hasn't been long), I haven't even thought about this. Yet another important concern that we need to address while we're still in beta.
If somebody asks a specific question rather than asking for a review, is that off-topic?
I'm going to start off with the "easy" response: no, because CR could be in danger of surviving if we're less of a Q&A site like the others.
The "harder" response: I'm still thinking no, as long as the question(s) is still on-topic. It may be a specific question, but it doesn't seem that specific. It still requires an examination of the code. It isn't a typical SO question that can (most-likely) be answered with a web search. Plus, with this specific question addressed, more reviews can be made. Yes, I still believe that, despite the scope of the question(s), answerers should be allowed to address anything in the post that may help the OP.
Should we accept any attempt to improve working code (which is more or
less what's actually happened) or should we only accept more strictly
code review requests (which is more along the lines of what the site's
FAQ and definition would state)?
Going back to my very first point: I don't think we should be restricted to only code review requests. Whatever the answerer has to address, as long as it's all relevant (and the OP's code is working), it should be allowed. I really don't think we need to be too strict about something like this..
We must also keep in mind that no one is obligated to give a full review. There are tons of examples of this. Plus, if one answerer primarily addresses everything or nearly everything (guilty as charged), then that may leave little to nothing for others to address. If the first answer has flaws or has missed some important things, then the additional answers can address those things. My main (hypocritical) concern with this is that it may keep our answer-per-question percentage low. However, we're not like the other SE sites: we don't just answer specific questions. But that doesn't mean that we need to change our ways outright. I think we should still be open to specific (on-topic) questions and/or code review requests.
tl;dr: "no" and "the former" :-)