Go to the school that fits you better for whatever reason. Don't let the name of the school throw you off.
You can look at some of my posts on other threads and see a little more of my story, but to put it in a nutshell, I started once upon a time at UT-Austin, and finished much later at UTSA. Because of my prior time at UT-Austin and because of its reputation, I was, at first, very concerned that my degree would lack the "name recognition", as you put it, if I couldn't go back to UT-Austin. But the HPO advisor at UTSA put it very succinctly one day when I inquired as to the number of graduates UTSA puts into medical school. He told me, "we put in as many students as are qualified." That was no BS. Work hard, and do the best you can in terms of GPA, MCAT, etc. A great application is a great application no matter the school from which you get your degree. Do not let the UT-Dallas name make you think you have less of a chance at medical school.
I, too, can attest that UT-D was well represented at all my interviews. And for what it's worth, UT-D is getting a lot more respect in terms of "name" nowadays.