The first sense, based as it is on a no-longer-common sense of beg, being a poor translation of the original terms, as well as not ever having been an adequate description of the fallacy, is not well understood except in specialized contexts, such as academic and legal argument.
The sense "sidestep or evade a question" is a simple development of the original sense that uses the expression to include any evasion rather than only one based on the logical fallacy.
The sense “raise a question, prompt a question” is more recent and has been proscribed by some commentators, but is now included without comment in some dictionaries. In the “raise the question” usage, it is used broadly, both for antecedents (given a conclusion or consequence, how did it arise?) and general questions (e.g., “Where do we go from here?”). Others suggest that the phrase is hard to understand in any event, and be avoided, using instead phrases such as “assume the conclusion” (for philosophical sense), "evade the question" for failure to address the question, and “raise the question”, “prompt the question” (for last sense).