Medium format is well established as 56x56mm which was commonly cropped at the easel to 56x42mm (645). Large format started at 4x5" or 9x12cm.
Well, any format that was cut from 120 size roll film was called medium format. 6x6 was not really any more established than 6x4,5 or 6x7, as far as I could tell. Similarly there were different sizes of large formats; 4x5, 5x7, 8x10, etc.
Why do we call 36x24mm full format?
When Canon first made the 1Ds they advertised it having a "35mm full frame sensor". The "35mm" was clearly present in the expression and people understood what it meant, it meant the sensor was the same size as a 35mm film frame. However, later people started to talk about full frame sensors and let out the format size indicator (35mm) which is kind of a silly omission. But often language use evolves in this way, the history is forgotten and terminology that is not really self-explaining sticks as common usage. As long as people understand what is being talked about, it's ok.
