In the context of your examples, either preposition could be used. The fact that they both come from different variants of the English language is not the reason for the difference.
As Independence Square is the proper noun of a place, the correct way to describe being there would be "I am in Independence Square". But the square itself is also an area of land, and when referred to as such you may see either preposition used.
Take the second example you gave about tents - we might say that someone has set up their tent in a field, yet we might say that a game of football takes place on a field.
The general rule for using in or on is:
IN - when something is located inside of a defined space, whether a flat space (like a square) or a three-dimensional space like a box.
ON - when something is touching the surface of something. It could be a horizontal surface, like a floor or beach, or a vertical surface, like a wall.
As a square (as in an open area) is both a surface and a defined space, either could be used. In example 2, the stage is both in the square (in the sense that it is set up within the defined area called 'the square') and it is also on the square (in the sense that it is set up on the surface of the square area.
So if your purpose is to state the location of something, you should say it is in the square, because you are using it as a place. If your purpose is to state precisely where something is and you have already defined the location (eg Independence Square), you might use "on" to indicate that it is actually on the ground. Note also that you could use "at" to state a location.
One possible reason for using "on" in the second example might be to emphasise the temporary nature of the tent city that had been set up there.