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Yes, O. engelmannii or a hybridized mixture thereof,...Meaning that in the Tucson area O. engelmannii and O. phaeacantha are the prominent Opuntia's there and those two species hybridize a lot with each other to varying degrees. The hybrid has been given the name "Opuntia x occidentalis, which is what I would say you have pictured. http://davesgarden.com/guides/pf/showimage/187744/
There are at least a dozen species of wild Opuntia growing in the Tucson area, but most books only cover a few of them, ignoring most and lumping their names together. While the various species do hybridize, the hybrids are not that abundant, and they are usually obvious among their parents. This is not to say that most species do not have hybridization in their ancestry, and this is a complicated subject. Regardless, most of the groups of plants that behave as distinct species in the Tucson area are not hard to recognize, once you get some practice. O. englemannii and O. phaeacantha do both occur in the area, and while not rare, neither is particularly common either. The first photo above, dated Dec. 17 at 4:22 is actually what is being called Opuntia orbiculata now (O. dillei is a synonym), and it is probably the most abundant of the larger Opuntia species in se. Arizona. Your second photo appears to include at least three species within the large clump that dominates the photo, but I can only identify the front left one with certainty (bluish pads and long spines), and that one is Opuntia confusa, a very common species in the Tucson area, and originally described from Tumamoc Hill. There are a few that the others could be, but the large one in the middle might be O. orbiculata, and the smaller one behind it to the right might be O. camanchica.
As for O. occidentalis, that is found in coastal southern California, and it does not occur in Arizona. It is a wild species that is closely related to O. engelmannii, and there is no good evidence that it is a hybrid.