Yesterday when I was solving some basic geometry problems, I was curious ( I'm still though! ) about something:
Why / When do some people indicate the angle with its measure $= 90^\circ$ by adding a point inside it?
I like history, too, so if it's possible I want to find the story behind this notation.

According to Wikipedia:
Note that it talks about an arc with a dot, not a square. For what it's worth, I've seen both "little square with dot" and "little square without dot" in Greece. I've never seen an "arc with dot".
I will be quite surprised if there is any particular story behind this notation. It is probably a matter of different communities using different notation for the same thing. Another example of this phenomenon concerns, e.g., the Notation for intervals.