After some tinkering with the editor, everything started working.
1. Removed my .pak file.
2. I created empty .xdb files that I needed (AdvMapObjectLink and Texture for in-game icons).
3. I filled them in manually using Notepad++.
4. I checked them after putting the .pak file back.
5. After checking, I put them back into the .pak file and deleted the originals from the data folder.
6. Everything works as it should. The icon is missing in the map editor, but that's not a big deal; I can fix it later or just ignore it.

Conclusion: It's easiest to use Hattabych's method. My method came about because I was too lazy to re-enter all the necessary elements. As you can see, this method is easy to mess up, although it can be used by people like me who already have a hero, but it was created manually, and I'm too lazy to redo it through the editor.

Laziness is the engine of progress. But in my case, it's an engine that needs to be repaired almost every step of the way and fueled with attention and effort. My method works, of course, but Hattabych is absolutely right, and his method works much better.

I haven't tested his method yet, but I'll take his word for it, because I don't really want to do anything more with new heroes at the moment.)

Added 1 minute ago
Now my hero can be placed on any map, regardless of the mod. As long as the .pak file is in the data folder, everything will work. The index.bin file is probably also needed, but I might be able to put it in the .pak file itself.

I'll work on that later.

Added 14 minutes ago
It seems that I can put the index.bin file into the .pak file with my hero without any problems, and everything works. Well, what can I say... I think I've figured out how it all works. I'll test the manual method of adding a new hero in a while. I don't think it will be easier than using the editor, but I want to experiment so that I can avoid fighting with the editor as much as possible if I want to add something manually again.