While both will kill Deer, you have to look at the shoots and take into account their age and size. Also legalities in your area. Therefore I would opt for the .357 of the ones given to select from. Just make sure you get a youth stock so the gun fits them, and they can shoot it well.
Now for other options, look at the .44 Mag. You can load .44 Spl to get a nice soft recoil, and still have some punch. Plenty for deer, or other thin skinned game. Then it is simply a matter of heavier loading to make it a Mag, that would work for bigger stuff. My wife has killed several Black Bear with her .44 Mag carbine. The .45 LC can pretty much do the same thing. My wife went with the 45-70 for our son when he was 12, he was small for his age at the time, but he handled the 45-70 well, liking it better than the .243 I had given him. Starting loads for the 45-70 Springfield worked well to start him out. Then as he grew, I raised the loads till we reached a good heavy load he likes. Now he loads with a 400gr bullet that rocks my world, I have all the confidence in the world with this load for any Bear in Alaska. I also would not hesitate to take it for Bison.