The "correct" use is unclear and not obvious. Even people who are aware of the correct usage frequently get it wrong (see comments here). There has to be a better term for this.
The "incorrect" use is actually useful (which, I guess, is why the phrase has been co-opted for it). There are statements that "beg the question" and that's a useful phrase to have (and a much clearer use of it). I shall continue using it for this :)
Begging the question means (to most people, I think) that there is an obvious question in response to the statement. The statement itself is "begging" for this question to be asked.
Raising the question means that one's response to the statement is to ask a question. It may not be obvious, and it's not an attribute of the statement itself that implies this response.
I think that's a useful distinction.
The English expressions I do wish people would get right are "moorish" meaning spicy (after the Moors), not "I want more of this", and "enormity" meaning horrible, not huge. But English is always changing, and using these phrases in their correct meaning now confuses people.