If your googling tells you that Square Dancing means Western Squares it is only telling part of the story. People who call and dance traditional squares would grind their teeth, and feel that their dance is being hijacked.
There are Eastern squares, and Southern squares too. I'm not an expert on which is which, but one form is more closely phrased with the music (i.e. do this in 2 bars, then that in 2). Some are obviously related to Irish set dances that evolved from the French C18th quadrilles. In the Irish versions each village produced their own set, slightly different from the rest and though the figures were fixed there was variation in that men could put in their own 'battering'(steps) at will (traditionally women never battered, they do now).
Western squares have been standardised so that wherever you go they are the same. New figures have been invented and given names so a short phrase with set the dancers off on an 8 bar or more sequence. To learn this you have to go to classes (basic, mainstream, plus and more). The other feature of Western squares in the UK is the dressing up in cowboy boots or frilly petticoats. You don't get that in people who dance traditional squares.
All sorts of squares are fun. Dancing in a set with other couples can give you a real high when it all works. And a good giggle when it falls apart and you grab a partner, 'square the set' and wait for the next turn of the dance to join in again.

Which is no reason for anyone to do it if it's not their thing. I'm sure you get as much satisfaction out of your dancing.