Snaffle bits are the term used for the large main ring of a bit; its usually the mouthpiece that is different. ie you can get straight mouthpieces ( are they called mullen?) single jointed, french link, ported, etc.
Trial and error I think as to which your horse likes. Not sure about the severity of them all, though I think the more joints the less servere?
You can get fixed rings, ie eggbutt, and loose ring. Loose rings are more flexible in the mouth.
Gag ( or bulbble bits
) have two or more rings below the snaffle ring,and one above; hanging cheeks have one above for poll pressure.
There are sooooo many bits its mind boggling
I'm sure someone will correct me if I'm wrong
I ride in a Jefferies revolver bit, lovely flexible mouth piece, its a gag, but I only ride on the snaffle ring