It's not just a question of context. It stems from a misunderstanding of the use of "beg" to mean "assume" instead of "ask for."
True. The mistake of using begging to mean assuming has been kicking around since the 16th century when "petitio principi" was mis-translated. Like you say, too late to do anything about it now.
Which begs the question - if it still works, why replace it?
It raises the question. Begging the question means something completely different.
No. Begging the question DOES mean raising the question.
The term "begging the question" originated in the 16th century as a mistranslation of Latin petitio principii ("assuming the initial point").[2] In modern vernacular usage, "to beg the question" sometimes also means "to raise the question"
Here's the thing: words and phrases can mean different things depending on the context. "Begs the question", when followed by a question means raises the question. "Begs the question" when talking about an argument means the obscure and antiquated English mis-translation of the older Latin mis-translation of the Greek phrase.
I suggest that you give it a rest. You're fighting the same losing battle that was fought over "gay" and "hacker". You won't change the public's mind, so the best outcome you'll ever get is looking like a pompous blow-hard. So, if that's what you're after, then have at it. Otherwise, learn to shut your trap and roll with it.
A penny saved is a penny to squander. -- Ambrose Bierce