Ken,
I am aware that in America, the word Twybil refers only to the smaller version, and the bisaigue is unknown.

Don,
The Twybil and Bisaigue do, in fact, have a German origin. The first tool along these lines to show up was the medieval German Kreuzaxt, or 'cross axe' which is basically identical to the later twybil. This tool developed in different regions into 3 distinct tools, the English twybil which was more or less the original, the French bisaigue which is more or less a gigantic version of it, and the German Stichaxt which did away with the mortising chisel end. These tools reflected the needs of the framers who used them, the stichaxt for example seems to have arisen around the time that the Germans switched to smaller timbers and shallow mortises. It is essentially a slick ideally suited to the German method. The bisaigue adapted to the large French joinery (in comparison to German) and from what I understand appeared in France around one of the times when they occupied the German-speaking Alsace.