Collounsbury, your post seems to suggest that you don’t know the meaning of “phoneme.” A phoneme is like a slot covering a certain range of phonetic space, which can accommodate a number of different “phones” (phonetic sounds). You described my analysis as “theoretical” — of course, the whole phenomenon of the phoneme is theoretical, in the sense that native speakers who haven’t studied linguistics are unlikely to be aware of it. Nonetheless, it is real just the same, as much as any scientifically defined phenomenon like the “isobar” on a weather map is real, even if people on the ground don’t “feel” an isobar when they feel the weather.
The wide ranges of variant vowel sounds you described are well known to students of Arabic like me. That was not in question. But I think you missed the point I was making, which is that, no matter how many sounds there are, they are all folded into only three phonemic slots.
Collounsbury, with your extensive experience in the world of social science, you’ve no doubt come across the emic/etic issue. Emic (from “phonemic”) is the way an outside researcher would look at a cultural phenomenon objectively, applying an academic frame of reference. Etic (from “phonetic”) is the way the people in that culture themselves look at the same thing from within their own homegrown frame of reference. There has been a lot of debate over which of these ways of looking should be privileged, or whether either should be privileged. I was going to make this the topic of another thread.
When we transcribe a character between slashes, that means it’s a phoneme. Between square brackets, it’s a phonetic sound.
So the phoneme /u/ could hypothetically take in any number of different phones, including [u}, [u:], [y], [U}, [o], [O], [@], etc. (Apologies for having to substitute ASCII characters for IPA).
Back to the OP, to restate my answer taking into account Collounsbury’s caveats:
The actual phonetic pronunciation of d.ammah, the Arabic /u/ phoneme in the name Usâmah, in the highest-profile dialects of Standard Arabic (Egyptian, Levantine, Iraqi, for example), is closer to the English vowel in push and pull than it is to any other phonetic sound known to English speakers. It’s definitely more [u} than [o]. The Arabs themselves might transcribe it either “o” or “u” indifferently, because they have only one phoneme to cover both sounds. So I wouldn’t advise taking their spelling as a precise guide to the actual sound.
As for the digraph “ou”, that’s a Frenchism, and the long history of Middle Eastern peoples mediating their languages through French when communicating with the West is a whole other topic for another day.