My family and I have always used "babcia" thingie, and I have never heard about "busia" thing before I joined this forum,
Neither have I. Never heard 'busia' in my family, but I could immediately judge it as having been derived from the word "babusia", the latter being perfectly imaginable, though equally unusual for me.
Does this mean that the Polish word for "grandmother" [babcia - Ziemowit] is a relatively recent invention and possibly dates from after the mass emigration to the USA at the start of the II RP?
It is much likely, in my view, although we must push back the time of mass emigration to the US to the times before the start of the II RP (end of the 19th and the beginning od the 20th century). As far as I remember, my grandparents (GF born in 1907, GM born in 1912) who lived in the country (south-eastern Masovia) and whom I visited quite often in my chilhood for a summer holiday would often refer to a grandmother as "babka" rather than "babcia", even if I myself used to address my grandmother with the word "babciu" [vocative of "babcia"]. Also, recalling the Polish historical films whose language was to some extent "styled" to sound less contemporary, I have the impression that the caracters in them would use "babko" rather than "babciu" when addressing the grandmother. The same can probably be found in the literature of the 19th century (someone may perhaps browse several "big" titles, those of Władysław Reymont or Bolesław Prus spring to mind, in search for that). So, all in all, one may perhaps assume that this diminution of the word babka to the word babcia started to spread out after the First World War. A similar process, the diminution of 'babka' to 'babusia' may have taken place in some specific groups (e.g. peasants of some regions) even before that time.
On a side note, my grandparents frequently used names of family that have faded away since then or are extremely rare now. Both my GF and GM carefully differentiated between 'ciotka', 'wujenka' or 'stryjenka', for example. I never dared to grasp the difference between those three, as in the 1970s and 1980s it was really unusual to hear the two latter terms among people in the towns.