On the 6V batteries (Sony NP-55 and another one) for my video light, there is a small switch (showing a red dot or not).
I could not find so far what this switch is doing. Who knows this?

It is NOT a switch, it is a small slider which you use to indicate to yourself the state of charge of the battery - for instance if you have several batteries and you charge them all up before going out on a shoot you would set the slider so that the red dot was not showing, then as you used batteries and they became discharged as you took them off the camera you would set the slider to show the red dot - that way when you get back home you can see which batteries need re-charging, - or indeed while shooting make sure you did not put a discharged battery back on your camera.

Thanks Beejaycee for your explanation.
That's why I could not find any difference in using or charging the battery with different switch positions.