how to correctly power the LED of a push button?
The datasheet you linked to covers multiple variations of the switch. I count 14 different part numbers. There are multiple voltages and colours.

The note under the table states there is a resistor included in the switch and special voltages can be made to order.

It is probable that the LED will light up on the 12v version if you apply 5v, it will just be dimmer - that may be acceptable.