The really correct usage is UCHI. I took a doctorate in economics (formal, not the business one) from that truly wonderful institution.
I had no problem with UChicago or “U.Chicago” in one hellish time teaching undergraduate economics with the unwashed, unprepared, and uninterested… no, really, are there such in economics?
Personally I dislike “UCHI.” It’s like an insider joke quite funny in 1880, but there is nobody left to understand and laugh. Ever try to fully explain a joke? Take it from me, just don’t try.
After all, Stanford is Stanford, Harvard is Harvard. The University of Oxford is “Oxford” or a worst “Oxf.” and with them the constituent school is often fully named, as in “Balliol, Oxf.” or “All Souls, Oxf.” For editors especially, when I read a published work that I care about, I like to see things like “Oxford Univ. Press,” because if I actually care, and might want to get the cit. or order the work, I’m not thinking of “OxfUPr,” something which a sincere scholarly publisher/author presumably wants me to at least consider.
In my PhD experience, I was supposed to be fully involved in “UCHI” graduate economics. For most of that time I was in fact flying in from D.C. or another coast as often as every week for 3-4 days and working the others. I thought, and still think, it should be “UORD,” ORD being the international airport code universally understood to signify Chicago.
That was a difficult but brilliantly rewarding program which I would do again in a heartbeat or those I have remaining to me, but all that travel… it could have been done with far less extra effort. Strange what we do when young, but then “Making Partner” in a big-big-time consulting firm seemed important then, equally strange now that I’ve done it, left it, and done it all again.