The L1 & L2 EVSEs do not "push" any amount of current in.
They simply advise the car that NN-amps (12, 16, etc.) is the Maximum that the car should attempt to "draw" (long term).
A particular EVSE might include a circuit to measure the out-going current, and open its relay if it thinks "too much" is being drawn. However, I think that is not a required EVSE function.