Exit wounds are good.
Ideally, the bullet would drop on the ground out the other side but that ain't the real world.
If your bullet would stop under the skin on a perfect broadside shot, it would lack penetration on a quartering shot.
Deer aren't hard to kill. The 1,000ft/lbs you put in the tree on the other side isn't needed.
The trouble here, and I'm not trying to be a jerk, is that you weren't sure what was beyond your target. While you knew from a pure safety perspective, you didn't know from a wounding and/or potentially legal perspective.
Your bullet did just fine.
No harm done, live and learn.