The dictionary lists both definitions as being correct, which is usually a sign that no one can agree on the true meaning of a word.

One seems just as intuitive as the other — etymologically, is there any reason why twice a month should be any more correct than every two months? The more you think about it, the more confused you might get. One of them will seem to make more sense at first, only for the other to gain favor when considered from another perspective (or after a heated debate with someone who disagrees). Is it more important to be technically correct by going with the original meaning or emphasizing the way that language evolves over time?