Difference between
"Hanging: Counter-cutting into his blade from below."
---Without going back to look at Meyer's Rappier text to clarify....., off the top of my head I would think this is referring to using a "Bogen." This is the equivalent of a "hanging guard" in later broadsword/saber methods, ie., as he is striking downward you lift your hilt and catch his blow on your long edge with the point aimed downwards out in front of you. It essentially travels from right to left.
and
"Taking out with the Short Edge- A short edge beat from left to right."
---This would be the equivalent of the Bolognese "Falso Parry", ie., point up and directed forward at the opponent as you parry from your left to right with the short edge.
These look similar as well as
"Slicing Off: Counter-cutting into his blade from above."
---This one is more of a deflective parry, somewhat like the inside or outside parry with the later broadsword/saber methods.
and
"Striking out with Hanging Blade- A high cut motion that strikes his blade out from left to right (beat from above)"
---This must be the "dempfhaw." It is more forceful than the slicing off and is meant to "ground" his blade, ie., knock it straight down from above to stop his motion completely and produce a "pause" prior to your counter. It is essentially a "trapping" motion. There is no equivalent in the later broadsword/saber methods.
---I hope that helps!

Keith P. Myers
Lifetime Member HEMA Alliance
Affiliate: Bartitsu Society & Cateran Society
Friend: Meyer Frei Fechter Guild
Rockville, Maryland