From everything I can see there is no engine here. This is a compressed gas motor. It works. It is a "steam motor" if you call Nitrogen gas "liquid Nitrogen steam". Note that the expansion cylinder of a water steam engine is the motor part. The "engine" includes that plus the furnace and boiler. This is my opinion of how those terms should be used, but there is no strict convention, AFAIK.
In this case, the Nitrogen in liquid form is stored mechanical energy like any other compressed gas except that in liquid form it is not under pressure. The pressure is generated as it boils. The recapture of the energy it took to liquify it is probably not achieved with decent efficiency. Also as mentioned, LN2 at atmospheric pressure, like this, boils off continuously.
Nice back yard project. Not an interesting idea, IMHO.