I was taught rugby was called rugby because it was invented in Rugby highschool
Yep that's right.
When the F.A. was formed in 1863 to come up with one agreed set of Football rules in England, it included Rugby rules delegates but they couldn't compromise and broke away from the F.A. in 1871 and Rugby remained a separate code of Football. The remaining F.A. delegates (Cambridge Rules and Sheffield Rules) were able to find a compromise between their two set of rules and created what we know as Soccer today - Association (rules) Football - in 1877. Rugby itself then split in two when those advocating professional Rugby (mostly the Rugby playing working classes of Northern England) broke away to form Rugby League in 1895. The same schism occurred in Australian Rugby in 1908. An interesting historical footnote is if Rugby League didn't form in Sydney in 1908 then the working classes of Sydney would've switched to Aussie Rules and the Australian sporting landscape wouldn't have ended divided by the so-called Barassi line.
As for Soccer, British sailors, merchants and migrants took the game with them around the world and formed clubs even in non-English speaking countries which the locals embraced and those clubs have now gone on to become some of the biggest football clubs in the world - eg. AC Milan where the AC stands for Associazione Calcio (Association kick/football) and Milan is the English spelling of the Italian Milano; River Plate which was formed by Argentinians who saw British saliors playing Soccer on the Buenos Aires docks; Athletic Bilbao in Spain is another that keeps its English roots in its name.