The expression "to beg the question" does not refer to a follow-up question that is required given an unsatisfactory reply to the original question. It is a debating term in which the question is the issue under discussion, and you beg it by taking for granted the very thing you are trying to prove, as part of your proof.
That said, no dictionary on Earth appears capable of a decent example of such a construction, and it could be argued that the "follow-up question" usage has the advantage of describing something that actually needs a description.
For example, dictionary.com suggests
Shopping now for a dress to wear to the ceremony is really begging the question - she hasn't been invited yet.Huh?
Collins English Dictionary merely defines the phrase as
(a) to evade the issue. (b) to assume the thing under examination as proved.
Those are neat definitions, but when would you actually assume the thing under examination as proved?
Another example might be the case where someone objects to same-sex relationships, then when asked why, says because it's wrong. There, the question is the wrongness or otherwise of such relationships. To use their wrongness as an argument is begging the question, because it's a circular argument that is only true so long as it's already true.