They in turn, probably got it from ARMA (the Association of Renaissance Martial Arts). You can read their essay on the subject here.
The thing is, they are specifically talking about 15th century and earlier techniques. The essay makes this clear in the first two paragraphs.
Many historical fencing enthusiasts do not grasp the concepts of parrying against cuts with cutting swords as described in numerous Medieval and Renaissance fighting manuals. These texts teach the concept of defending by counter-striking or by receiving blows on the flat portion of the blade. As will become clear, edge-on-edge parrying was not taught as doctrine. In fact, defense, or warding of cutting blows, is described in many ways in 15th century fencing texts by many masters and never as a direct resistant block of deliberate opposition of sharp edge on sharp edge (so common in stage-combat and sport fencing and derived from 18th and 19th century methods of swordplay).
There is a tremendous, if not outright complete, lack of any support for doing so that can be found within any of the source literature (at least prior to the 17th century).Since the GAoP starts in the last quarter of the 17th century, it is clearly outside of ARMA's essay. Fencing manuals from the early 17th century clearly show edge on edge parries. Here are some (rather explicit) examples.
Not only is the edge shown facing the opposing blade but the wrist is straight.
So, combat styles changed between the 15th and the 17th centuries. This should not come as a surprise. Everything else changed. The 15th century swords were big heavy bars of iron. They were used two-handed, often against an armored enemy. By the 17th century, guns made armor too heavy to wear. Swords got lighter and the steel they were made from got better.
What about marks on surviving swords? A few points here. I have done edge on edge combat with real swords as well as schlager blades which are close to real weight. A tempered steel blade does not show much damage. Blades that have not been tempered do show significant notching. Those are the ones that would not have survived. Remember that only a fraction of swords have survived, mainly dress swords that never got near actual combat.
Even when an edge is notched, it isn't that hard to fix. I have had to take notches out of my pruning shears. You can get rid of most of the damage quickly with a hammer and anvil.
A final note - the Mythbusters declared it a myth that you can cut one sword with another. They did it several times but they disqualified these because the sword broke instead of being cut. I'm not sure I agree with their assessment. If I hit someone's sword and it separates into two pieces, I don't really care if I cut it or broke it.