If it helps, one way to rephrase the original sense of "begging the question" was "asking to have as an assumption, the thing that is in question".
So for example, if you were to argue something like "Chemical X will give you cancer because it's a carcinogen". The evidence that you're offering, "chemical X is a carcinogen", is just a differently worded version of your conclusion, "chemical X causes cancer". You're asking (begging) for "it's a carcinogen" to be taken as a true fact and entered as supporting evidence, when the question you're supposed to be answering is whether that's true in the first place.
So that's why it was once called "begging the question", but the twist of logic it takes to understand it is a bit far to reach for how we use the words now. Which is why it would probably be best to abandon it - find a new term that more obviously describes the logical fallacy and let "begging the question" mean "begs for us to ask the question", in the way people are already casually using it.