I don't know why people, mostly DOs, make such a big deal about there being two degrees. I was accepted by an MD school and decided that I would rather go to the DO school that I was also accepted by. My reason was that the MD school was still very new and I would have been in the third graduating class, but the DO school is much more established, has much better rotations and residencies, has more research, and better academics. I am sure the MD school that accepted me will bypass the DO school in research but it won't be happening while I am there. I also choose the DO school because I know I will have more opportunities for residency positions since I can compete for both DO and MD residency slots and neither MD nor foreign educated physicians can apply for DO residencies. I didn't choose the DO school because of OMM, I do not have enough OMM experience to decide what medical school I want to go to based on that, I made my decision between the two school based on which I believe would train me to be the best physician. There were MD schools that I didn't get into that if I would have been accepted I would have choose them over the DO school.
As far as the hospitals not distinguishing between the two, that is completely ridiculous. If you are an MD you should be distinguished as one, and if you are a DO you should be distinguished as a DO. I would be upset with my hospital if I was a DO and they gave me a badge that said I was an MD. This isn't because I think I am better than my MD colleagues, it's simply because I didn't earn a MD degree, I earned a DO degree and calling myself an MD would be misleading. The people who don't care about the hospital mis-identifying them either don't care about slightly misleading people or really want to be an MD and finally have an excuse to act like they are. Personally I would either request to be correctly identified as a DO or I would just white out the MD and use a sharpie to wright DO. Again this is not out of malice towards MD's, it simply is correcting a clerical error.