Steve,
My wife has a painting lesson with a gal from town who is a botanist. She already has the question to ask her! I looked it up and believe the one we have here is known as "optunia leptocaulis" which might be a form of the tree cholla. The reason for the thin look is that we don't get much rain here. 7 inches of precip per year is not much. Most of the time recently we have had less, however, so far this year we are doing very well.
Al