It's thought that a homoygous grey (GG) will turn white without fleabites, and a heterozygous grey (Gg) will go fleabitten.
This has to be a wrong assumption as I have had many snow white mares out of bay, brown, black and sorrel mares. I have a white Playgun granddaughter out of a bay mare right now. Playgun, himself, is snow white and is sired by the Sorrel Freckles Playboy but is out of a snow white mare. She produced another gray son by a sorrel stud that is also snow white. They cannot be homozygous for gray with only one gray parent.
I do not think there are two 'completely different' gray genes, but I would bet on one have a slight mutation from the other. I have never seen the two different patterns ever cross the genetic lines. They have always had a direct line to one dominant gray pattern or the other or it is a 'crap shoot' if they can have both from two different pattered parents. I have never seen either pattern skip a generation so it is dominant trait.
The same is true of the Arabian, Nabor*. I had two different Nabor* daughters I trained out of bay and chestnut mares and both were snow white like him.