The term AFPS is wrong. As soon as people shoot each other in a computer game its an arena. Or is there a possibility to go out of that arena and do something else like not being shot?

What they might wanted to express with arena shooter is proper level design shooter. Proper in the way that the gameplay itself and the corresponding possibilities when designing a map are directed to give the player an interesting challenge.

In contrast to realistic looking level design shooter. Where the gameplay is realistic and thus the level designers have to live with only limited possibilities for giving the player an interesting challenge.

How close to reality is it to control a mouse and a keyboard and while watching an interactive movie? As opposed to using your actual legs to move.

Thus my proposal for a new term is at first, do not give proper shooters a different name. Call it FPS. Name the shooters that limit themselves to imitate patterns found in the real world a different name. Call them Realistic FPS. RFPS.

Call of Duty - RFPS
Battlefield - RFPS