It's not the Red Light. It's the CLICK!
Well the click can be a problem as well - but as someone that started recording long before using clicks was commonplace - choking from the awareness of being recorded can be a real problem for many. It was for me.
As Bermuda mentioned, it is just like stage fright. That voice in the head that starts panicking over concern of this performance being scrutinized, that it will (good or bad) last forever... that fears wells up eroding first confidence, then concentration, focus, coordination...
What to do about? IMO just like any fear - realize it is all a mind thing... which likely means confronting over and over, trying different things in one's head to conquer it... or at least work around it.
In the past, I remember attempting to convince myself to simply not care (not care if it goes well, not care if I get the job or not, etc.), I remember sometimes relying on the old Dune trilogy mantra "Fear is the mind killer" - basically any thought or technique one can use to bolster confidence.
Certainly being prepared, being able to play well helps - but it's not ultimately the solution - if our mind/emotions refuse to play along.
In a nutshell - you just have to confront it, over and over again. Just like many of us had to do with getting used to playing in front of people.