Magnesium air-fuel cell or battery?
What you have is classified as a Mg-air battery. You can read more about it here : https://en.wikipedia.org/wiki/Category:Metal-air_batteries

These are a class of primary(i.e. not rechargeable) batteries, where the anodic reaction is a metal being oxidised and the cathodic reaction is oxygen reduction.

The reactions are:

$\ce{Mg → Mg^{2+} + 2 e^{-}}$ on the anode

$\ce{O2 + 4 e^{-} + 4 H+ → 2 H2O }$ on the cathode, if the reduction goes to completion. Most of the time, it is only reduced to the peroxide or the superoxide which is considered incomplete reduction.

For it to be classified as a fuel cell, the fuel for the anodic reaction would need be brought from the outside(source: anectodal, personal thought).