Ugh, I kind of hate the internet's hardon for katanas.

On the cutting edge, he's not strictly wrong but he's not correct either. A curved blade means a smaller cutting surface which focuses the pressure of the blow onto a smaller area, making it easier to cut through something. It's the same principal as a musket ball vs a modern bullet; a modern bullet will make a much neater hole through you because the kinetic energy is focused on a much smaller point of impact whereas a musket ball is more broadly focused and will simply smash its way through whatever is in front of it. He misses this also when he starts talking about rapiers as well.

As to his comments about different varieties of cuts, I can't really speak to as I'm not an expert in the historical martial arts.

He was essentially right in the opening part of the video; there is no such thing as a "best" weapon. All weapons were made to fulfill a role and do a job. A katana was designed with slashing in mind whereas something like a European hand-and-a-half sword was optimized more for stabbing.

These forms generally are dictated by the type of protection your opponent is wearing. A curved blade is good when your opponent has lighter or less armor but wouldn't be as useful against someone wearing a hard, heavy armor like solid plate. If you look at the design of weapons from a given area or time period you can get a general idea of the types of armor that was commonly used.

If we look at someplace like the Middle East you have high temperatures and widespread use of horses or camels in combat as well as mounted archery which requires a great deal of flexibility. These conditions are not conducive to using heavy plate effectively and so it wasn't employed widely. Consequently a curved weapon would have been better to use as it could cut through lighter or lesser armored opponents because of the curvature of the blade.

These are not laws of nature but general trends that tend to show up in the history of war.

There's been a couple threads about katana's versus other swords. Here's one and here's another Hope they help.

That video is much too obnoxious to watch, sorry.

The short answer is that katanas are a slashing sword for use as a sidearm against unarmoured opponents. Their use against armour is limited. The lack of a pommel and crossguard limits the weapon's use in close-quarters combat and limits its defensive potential. Their thrusting potential is limited.

The longsword (the katanas contemporary in form and function in the West) was a combined cut and thrust sword that gradually developed into a primarily-thrusting weapon for use against heavy armour. It's heavy pommel and long crossguard were potent parts of the weapon - the crossguard was useful defensively, and both were powerful offensive weapons. It was excellent at thrusting (which is required against armour), and was often used half-sword against an armoured opponent.

The longsword was also primarily a sidearm, as spears and poleaxes were the preferred weapon of footmen at the time.

Did you have specific questions?