(From Wikipedia) To "beg the question" (also called petitio principii) is to attempt to support a claim with a premise that itself restates or presupposes the claim.[6] It is an attempt to prove a proposition while simultaneously taking the proposition for granted.
When the fallacy involves only a single variable, it is sometimes called a hysteron proteron[7][8][9] (Greek for "later earlier"), a rhetorical device, as in the statement:
"Opium induces sleep because it has a soporific quality."[10]This seems wrong: The Oxford dictionary lists a) the original meaning and b) "invites the question", and b) is not marked as vernacular.
Based on the context, you know exactly which meaning of the phrase is intended here, and it's disengenuous to pretend otherwise. The differences in context between the two meanings make this a non-problem.
Personally speaking, it's not always easy to tell which "begging the question" someone means from context. I often have to stop and ask "hold on, do they mean the premise assumes the conclusion?" (And all the worse for me if they just mean "brings up the question", because I'll spend a minute making sure I'm not just missing the fallacy).
In that sense, it's not quite the same as "slippery slope," which doesn't get used literally much anyway.
I'd imagine that we come to know the meaning of >99% of words and expressions through usage, so it feels like a fairly expected -- and reasonable, yet wrong -- outcome.
Now the people routinely use a different meaning, our language is unnecessarily muddier.