As I understand it, that's not actually correct. The entirety of the OT period is treated like the end of the half or end of regulation. They would start the 2nd OT exactly like the first OT except I think they switch to whoever lost the coin flip in the first OT. Maybe the rules change that, but I'm pretty sure that's how it was previously.
No, they said it over and over, even Vinovich said "We're starting a new game." Thats why there was no 2:00 warning for the OT period, because it was like any other 1st qtr.