What all did you install BEFORE 7th heaven?
Like I said, nothing. I made a fresh install of the game on the C: drive to a non-system/users folder. Then I installed 7H with the game converter etc. and ran some mods.
Okay, thank you everyone! I deactivated all the mods (Menu Overhaul, Field Textures, music and weapon models) except the NT mod and started a New Game because I have indeed loaded from a pre-existing save file like Sega Chief said. The mod works correctly, I guess. No triple AP weapon for Barret this time and Cloud has 3 different materia. (Strangely he just had 2, bolt and ice when I first played the NT mod). It seems there are some mods who mix up parts of the kernel, but I'm not sure. I'm going to test this with the Menu Overhaul again.
Well, at least there are no triple AP weapons anymore. They definitely ruined the beginning of the game and made the bosses far too easy.