How to integrate hardware/embedded button with LEGO?
For the first switch I bent the legs of the button and splayed them out just a little bit.

Next I stripped off the caps of two Dupont connectors, ran them through the headlight bricks and connected the button upside down.

An axle will trigger the button. Either one with a stud on one end (and a 1x1 tile) or a regular one with a half bush.

A 2x2 brick on top serves as a button.

The second switch is a hacked light brick. They're very inexpensive on bricklink.

I took out the batteries and the LED (they're not soldered as they're meant to come apart) and replaced the one on the right with a metal contact.
For that I took a paper fastener, cut off one of its legs and bent it in an e shape.