that is not done in the Netherlands. For cyclists you just point in the direction you're going with the arm closest to that direction.This is taught to children at a young age.

For driving a car I was never taught hand signals, I suspect that if you drive while your lights aren't working, you'll get a fine if caught. Wikipedia says that when driving oldtimers, which often don't have a cab, you just point in the direction you're turning, same as on a bike.

That is definitely not universal, as in my country you just use the arm on the side you are going when on a bike.

That's not universal at all and completely wrong in large parts of the world.

In all the Nordic countries, who do ride bicycles very much, and in much of Europe, the right arm is used for signalling right turns, and the left arm for left turns and stopping/slowing. The stop sign is almost what you describe as a right turn: left arm up as an L and open palm. The left arm is used for stop, because it's most visible when riding/driving on the right side.

Left turns are mostly used on small roads, because in larger intersections the cyclist should preferably make an "orientation stop" to make two straight crossings instead.

This signalling system was introduced many years ago when most bikes had pedal brakes, so there were no issues in braking while signalling with either arm. It can be slightly annoying for bikes with hand brakes only, since the front brake is on the left (which is also not a universal design). But it works, people do use it, and they do expect others to use it when it makes a difference. For instance, it would be considered rude to make a right turn without signalling if there are cars waiting to turn right, because the bicycle has the right of way it were to continue straight.

It's also not the same for mopeds and motorcycles, because they should always use indicator lights, but in case it doesn't have those and since the accelerator and front brake are on the right hand, they can only use the left arm to signal in those rare situations.

The only times I've seen cars using hand signals have been at veteran shows where the passenger would do the right turn signal.

I've never seen or heard of this. Interesting.

Well in my country you point to whichever side you're going.

Not universally incorrect. There seem to be regional differences. In Germany it is: use the arm on the side you want to turn to.

This is what is done in Canada

Those are the correct signals in the US, but most drivers don't know them.

Pointing with the arm nearest to the turn is very clear.