Edit: Upon further investigation, I have realized that there are more listings of runs on Mega Man RTA leaderboards. I wish there were more guides here because of situations like this.

I'm currently watching Chapter 2 and he does not have an MKIII (unsure what the name is since I'm playing in US), but it magically appears in his inventory. What is this? I'm trying to get notes written up so I can learn this. Where do I get this MKIII weapon? I've watched his movement and wrote it all down.

You've just noticed the most important trick in the MMXCM glitched runs. I'm not sure if he has an FAQ to explained it, but anyway I'll leave a short answer here.

In japanese version (PS2/GC) of MMXCM, the first slot in Weapon and the 101th (last) slot in FM list actually points to the same memory address - that means you can do the following: (1) buy a specific weapon, put it into the first slot (2) manipulate the FM list manually, and move the cursor to the last of the FM list (3) you will find "the FM corresponding to that weapon" in that slot. and vice versa - buy an FM and convert it to a weapon.

In chapter2, if you buy a thunder shield, then you can convert it to an X-buster MK3 using this glitch. Auchgard has summarized a table for GC version (some of them leads to different result in PS2 version.) https://pastebin.com/5pPvG2MZ

I've also record a few clips for this glitch, although I only write the explanation in Traditional chinese.... but anyway it is much slower than a speedrun, so I guess it will be easier to learn that trick. https://www.youtube.com/playlist?list=PLGWAiw0g_CJp7FkN33qxd-2J-D-88hH2B