To my way of thinking, any rifle that I can hunt with is a hunting rifle. It’s not limited to only “big game” but to all game and to varmints too.
I have used;
40 cal flintlock.
50 cal flintlocks,
58 cal Hawken cap-lock
62 cal flintlocks.
6.8 SPC AR-15s
I have also hunted with and killed with the following handguns
And I add to all the above list 7 kills with recurve and long bows 5 deer, 1 antelope and one elk.
2 of the deer were killed with aluminum arrows and all the rest of the animals were killed with wood arrows. None with compound bows, but I never owned one of those. If I did I am sure I could have used it too.
Hunting is a skill.
A “hunting rifle” is any rifle that can be used while performing that skill.