If you want to describe that specific fallacy in some way, there's a lovely phrase for that which already exists, describes the fallacy better, and uses fewer words - "petitio principii".
"Beg the question" used in this sense strikes me as redundant, inaccurate as a translation and a description, clunky, and pointlessly confusing. I hope its misuse is not actively fought, and that people who wish to describe the fallacy find a better way of doing so.
Even better, it is impossible to confuse with the "proper" usage (because it is distinct syntactically) and makes the "proper" usage more transparent and easier to understand.
The everyday form is transitive, its always "<foo> raises the question <bar>" and means "Accepting <foo> creates the need for a resolution of the issue <bar>".
The "proper" form is intransitive, its always simply "<foo> raises the question", meaning "<foo> invokes the petitio principii fallacy". This can be rephrased, however, as "<foo> creates the need for a resolution of the issue <bar>" where <bar> is the proposition argument <foo> was offered to justify, as it is stating that the argument depends on the conclusion it was offered to support.
http://en.wikipedia.org/wiki/Mondegreen
Actually, reading the article, it seems to make the distinction that those would be malapropisms instead:
Literally.
Nice examples. I also regularly see "reigning in" when "reining in" is what's meant. Reigning is what a monarch does to a kingdom, reining is what a cowboy does to a horse.