Switch – How Do I Program this Timer Switch

You say that, when you release the button, the contact opens again (resistance goes to infinity). But the light stays on, right? That means there must be some device which has a "memory" of the last action: you push a button, the light goes on and stays on, until you press the button again (or another button). I'm sure there is a device in the cabinet (Dutch: "schakelkast") which toggles its output on every pulse it receives, and which in this way controls the lamp.
The wire on the top right of the switch also hints in this direction; it's a signal wire, it's too thin to be a power wire. My hunch is that it carries the pulses to the actual switching device (I don't know what it's called).

If you want a single control to do this, look at multi-position rotary or slide switches. It's fairly common to have a switch that connects a common to two outputs (let's call them A and B) and gives you the choices off, A only, A and B, or B only. Variants which drop one of those options are also readily available.

Switches of this sort are often seen in multi-bulb table lamps that offer several settings. They used to be common in stereo components to select which of two sets of speakers will be powered. They're also used in slightly modified form as the pull-chain switches that control ceiling fans.

A good electronics supply house should have several alternatives which meet your needs, differing in size, in how they're mounted, in the kind of knob they have (or the shaft to which you can attach a standard knob), how much voltage and current they can tolerate, and so on. You may still be able to find suitable switches at Radio Shlock, though they've phased out most of their hobbyist-friendly parts department.