Right, except you have it backwards and want to use the COM and NO terminals. “Normally” in this case means while the switch is unactivated, i.e. plunger out, which is the state when the door is open. You want the switch to be open in that case, and closed when the plunger is depressed.
As best I can tell, the one you’re replacing is a NO SPST momentary contact switch - normally open single-pole single-throw switch that is closed only while the plunger is depressed. The one you’ve ordered is a SPDT momentary contact switch - single-pole double-throw. There is no functional difference between your old switch and the new one using the COM and NO terminals. EXCEPT!!! The NC terminal may be energized while the door is open. It needn’t be - if the unswitched lead is connected to the NO terminal and the switched lead is connected to the COM terminal, the NC terminal will never be energized. But you may not know which lead is which and get it the other way about. To be safe, ensure the NC terminal is protected from making contact with anything. An insulated female spade connector on that lug with no wire attached should do the trick.