You've got it in a nutshell essentially. The middle piece allows the bit to better follow the natural contour of the mouth rather than to bend into an angular "A" shape and risk poking the roof of the horse's mouth or pinching his tongue (ouch).
Actually, the fact that it is thicker is also more beneficial to the horse's comfort. A narrow bit like the one you pictured applies more pressure over a very small part of the horse's gum, "cutting" into it and likely causing more discomfort.
Additionally, the link in the center of the lower bit pictured is often said to encourage a horse to chew on the bit and play with it in their mouth to some extent. The bit looks as though it is made of copper or another metal alloy which are sometimes termed sweet metal bits and are meant to stimulate salivation. All of this salivating, chewing, and tongue action acts to futher relax a horse and keep him soft on the bit rather than to become rigid, stiff, and resistant.
As you may be able to tell, I am a strong proponent for the linked bit. ^_^ I use one with my pony and it has definitely helped with her trainining. She has become much more responsive and light. Not all of this can be credited solely to the bit (it took a lot of work on my part) but it certainly helped. Bits made by HS Sprenger are my absolute favorite... though a bit on the pricey side. Below is a link to the bit I use with my pony: Herm Sprenger KK Ultra Loose Ring Horse Bit - Dover Saddlery