To be fair, the phrase "begging the question" makes almost no sense from a modern English perspective- according to Wikipedia, it's already a bad translation of a Latin phrase that's tied pretty closely to a specific debate format.

By contrast, the colloquial use feels like an abbreviation of the implicit phrase "it begs for the question to be asked", which makes so much more sense than the "correct" meaning that if I'm being perfectly honest, I'd rather use it.

I like Wikipedia's alternate name for the fallacy: "assuming the conclusion", because it explains what's actually happening.