@John - The people growing wolfberry here that I know have used Lycium fremontii (also native). I personally don't have it on my property (although I would love to) due to a fairly severe vision disability. I try to limit my thorny plants to the overstory so I don't bumble into them as that never turns out well. Bit challenging avoiding thorny plants in the desert though.
Now that you mention it - I wonder if sometimes Lycium fremontii doesn't get confused with Lycium exsertum. According to
http://www.desertmuseumdigitallibrary.org/public/detail.php?id=ASDM01419 we have 15 species of Lycium in the Sonoran desert.
Great question - thanks for asking it.
Oh, and to answer your question about Chinese Wolfberry (Lycium chinense), I've heard of people around here trying it but I haven't followed up with them on their success. Mostly I prefer to grow natives if there's a suitable native alternative, but I certainly have plenty of exotics too.