I am talking about pedestrian crossing traffic lights, that can be called by pedestrians to turn green. Appears that these are indeed called beg buttons.

They are called like that because they don't actually do anything. They are there just to calm down impatient pederastians by giving them illusion of control ;)

Not universally true. I have seem some crossings in Texas and elsewhere where the light is solely for pedestrian crossing and will change (after a short delay) on command. On intersections that are sensor-controlled they will also change the light if there is no cross traffic to change it for you.

That said, on intersections where the lights are purely timed or where there are always cars waiting in both directions, it pretty much is a "beg button". At that point, there is no way to allow on-demand pedestrian crossing without making the roads useless to cars. In fact, some places like NYC don't even have buttons because they are pointless and possibly counterproductive.

That's largely a myth, at least in the US. Everywhere I've lived really needs a button or the "Walk" cycle is skipped to give longer turning times for cars. But because of the myth, you get infrequent pedestrians who believe in the myth and just wait until somebody on either side actually pushes the button.