Is this product description wrong and a simple battery and not a fuel cell?
Your analysis seems correct, i.e. it is only a fuel cell if the magnesium is being oxidized by the air. You can prove if that is the case, and do so as part of your classroom demonstration, by excluding any air, and therefore oxygen, from the fuel cell.


Use distilled water and fairly pure $\ce{NaCl}$ (rather than table salt) to make the salt solution.
Boil the salt water for a minute or so to remove air bubbles (some oxygen from air dissolves into water).
Put the cell with electrolyte in a plastic bag sealed around the wires and a narrow tube such as a plastic coffee stirrer or soda straw that you'll use later to admit air, carefully squeezing out air bubbles. Seal both bag and tube with plasticine modeling clay or soft wax.
Observe the open circuit voltage of the cell.
Then run it for a while with a load (e.g. a 75 mA 1.5 V lamp) or a 100 ohm resistor, which will draw 1.3 mA), depending on the current output of the cell. You would expect some voltage drop over time, either due to cell polarization or due to any residual oxygen being used up.
To prove that this is a true fuel cell oxidizing the magnesium using air, bubble some air into the plastic bag through the (unplugged) tube. *If the voltage goes back up, then drops down until yet more air is added, you've shown that oxygen is being used and it's a true fuel cell.


Please accept my compliments for your thoughtful question; do post your findings here!