The second of your meanings is closer to the original meaning. Begging the question in that sense involves unjustifiably claiming, via faulty, circular reasoning, that the conclusion is true (though, of course, it may well be). The first of your meanings is a modern, regrettable, phenomenon. It could be that "beg the question" comes from "beggar the question," which means to render the question pointless as the conclusion (the answer to the question) has already been assumed via faulty reasoning to be true.