BLUF: I'm pretty sure the answer is send the note anyway but would you change the contents of the note in some way?
I've had this situation several times and I'm not sure what to do. Having High C tendencies I have been locking up. The situation goes like this: Set up the interview through recruiter or internal HR. Have the interview and it goes well. Sometimes before I can get notes even written I get contact back saying they want to set up the next round.
In the area where I live and the sector I'm in the market is very hot. The unemployment rate is low. So when a hiring manager finds someone they like, they move fast.
Should I thank them in the note for moving on to the next round? Do I do that instead of referencing something that we talked about where we agreed? Does any of this change when I'm working through a recruiting firm instead of applying myself? Usually I talk to the recruiter right afterwards to give my feedback and tell them how the interview went and express my interest so they're already getting that feedback through the recruiter.