The second one is a french link loose ring, my personal favorite, the first is a tom thumb, commonly called a tom thumb snaffle, but it is a shank bit with a jointed mouth. I would try her in the french link and see how she does.
Use the second one posted! That is called a french link snaffle and is very nice. I'll still send you the bit I said I would in case your horse doesn't like the french link (doubtful).
Steer clear of the first bit you posted though.
As everyone already said, the 2nd one is a snaffle. The Tom Thumb is often called a "Cowboy Snaffle" but it is in no way a snaffle. A Tom Thumb (American style) has a straight, short, shank, and is used with a curb chain. It is a rather unbalanced leverage bit. There is nothing wrong with a leverage bit if your horse neck reins and you ride with very loose contact, but most horses really don't need them.