How many bits are in the snaffle family?

Technically, a snaffle bit is any bit without a shank. Any bit that creates direct pressure without leverage on the mouth is a snaffle bit. The most common snaffles are the simple loose or D-ring single jointed ones but double jointed and mullen mouth bits without shanks are technically snaffles as well.
Pelhams and kimberwicks have shanks so they aren't considered snaffles even if they are jointed. Curb bits are shanked so they are not snaffles either.

There are two components of snaffle bits, the rings and the mouth piece. Snaffles can be almost any combination of the two.

Some types of rings on snaffles are:
Full-Cheek
Half-Cheek
Loose Ring
D-Ring
Eggbutt
Baucher (hanging)
Fulmer

Some types of mouth of snaffles are:
Mullen Mouth- A straight bar
Single or Double jointed
Dr. Bristol
French Mouth
Twisted
Corkscrew
Single or double wire twisted
Hollow
Roller mouth

Since there are many combinations of rings and mouth pieces, there are many types of snaffles! It would be difficult to name them all.