The 13th century was a transitional period in arms and armor in western Europe. The Oakeshott Type XIIIa "Gran Espees" do start appearing, and mark a shift from one handed arming sword to two handed longsword. But they tend to be beefier with one or more fullers rather than a central ridge and a less defined point. Since the earliest detailed treatise on sword fighting from that period is I.33 in around 1300-1320, we don't have anything directly relating to the use of Type XIIIa's. But it's likely they were used with covering blows (due to their length and crossguard) and may have been used with the early versions of the longsword techniques as explained in the treatises from subsequent centuries. We know the Liechtenauer sources refer to a long-existing system of longsword fighting. And yes Albion does make some very nice versions of these swords as well. They feel more blade-heavy in the hand but are still nimble. It's possible they would have been able to cut through a simple gambeson, but I think any decent mail or early plate would have been proof against a cut from one. So personally I suspect that the dominance of the cut would have been for clearing the center of the fight and overmatching the opponent's blade and shield/buckler rather than actually trying to hack through heavy armor. Albion also makes a special sparring sword in that style, and it would be interesting to pair it with Jake Norwood's theories on the "common fencer" methods that may have pre-existed Liechtenauer's advanced system.