If you go the DSLR route, and know that you don't need super low light performance (i.e. not a full frame) then choice of body is down purely to ergronomics. Once you choose your sensor size in the DSLR world, funcionally there is little real difference between bodies and imo it's obsessive camera nerdery to argue otherwise.
I'm currently running the smallest body that Canon makes (100D) with a fairly wide selection of prime and zoom lenses (including the standard camera buff telephoto and wide angle L zooms). I had considered a 6D for low light performance, but that body was so ridiculously heavy and at four times the price I realised that it would have spent most of its time as an expensive paperweight. I have a kid, and these days instead of toting camera and sports gear around, I now carry around kids gear.
Overall though, I tend to agree with the others who are questioning your decision to go with a DSLR at all. Only go with a DSLR if you intend to buy and use a selection of very different lenses. Otherwise stick with a point and shoot or mirrorless body.
A camera will not lead to better photos. Knowing how to use any camera will lead to better photos. Well, I'd draw the line at smartphone cameras, which tend to be a bit sucky most of the time. For example, almost no camera will give you good night time photos handheld (well, maybe a 5MIII with a 1.2 prime, which is many times your budget). But a cheap DSLR with any prime and and a tripod and remote release will give you great results- you just need to know to carry this stuff and how to use it properly.