It's hard to say, could have been fungal or bacterial infection, sometimes they can look quite similar. I would assume (in my personal experiences and opinion) it was due to the infrequent water changes just because fungus and bacteria can come about with decaying food and fecal matter and remember the tank temperature is warm as well. The tank you had was a little too small to leave unchanged for 2 weeks. However, this is an honest mistake! I wouldn't feel bad about it just because it may not have been exactly that, it's usually a combination of factors. I did the same thing once with one of my old red female's tank
she got a similar white covering around the gills
her name was Ruby
Did you acclimate him to the new tank temperature when you added the heater? sometimes that can cause stress, lowering the immune system making them more prone to infections.
as for the grey body, that seems normal, along with a thick mucus coat around the body as well.
I hope you feel better about your fish
losing a betta buddy is never fun...RIP little guy! I am sorry for your loss