Fun fact, I was an early and enthusiastic user of Zimride (for making a long distance relationship work between the Bay Area and SoCal). I liked the product and once upon a time drafted an e-mail applying for a position there. I assume I would have been employee like ... 5 or something. Never sent it and it hung out in my Gmail for many years. Decisions, decisions :-)
To me, the key change deserving a new term was allowing completely independent, non-professional drivers, and I think Lyft beat Uber to that. (Heck, some purists would say "ride sharing" should be reserved for cases where the driver doesn't make an wages, just reimbursement for costs.)
Uber launched UberX in June 2012. Lyft launched in June 2012.
Give Sidecar credit where it's due.