In the context of LW, all those terms are pretty closely related unless some more specific context makes it clear that they're not. X-rationality is a term coined to distinguish the LW methodology (which is too complicated to describe in a paragraph, but the tagline on the front page does a decent job) from rationality in the colloquial sense, which is a much fuzzier set of concepts; when someone talks about "rationality" here, though, they usually mean the former and not the latter. This is the post where the term originates, I believe.

A "rationalist" as commonly used in LW is one who pursues (and ideally attempts to improve on) some approximation of LW methodology. "Aspiring rationalist" seems to be the preferred term among some segments of the userbase, but it hasn't achieved fixation yet. Personally, I try to avoid both.

A "Bayesian rationalist" is simply a LW-style rationalist as defined above, but the qualification usually indicates that some contrast is intended. A contrast with rationalism in the philosophical sense is probably the most likely; that's quite different and in some ways mutually exclusive with LW epistemology, which is generally closer to philosophical empiricism.