If I understand waht you are saying is you hot swapped the display from one machine to the other. In other words both machnes were already up when you unplugged one and plugged into the other.
Normally you should be able to do that although you may need to fiddle with the Displays preferences a bit (Detect Display for example). But apparently that wasn't working for this display.
Booting up with the display already connected should certainly work. And if I understand you it did.
The Mac Mini was off when I hooked up the display, not on.
I turned off the iMac when re-plugging the display back in.
I did fiddle with display prefs as first thing. No impact because the prefs *showed* that the Cinema HD was there and working! (Just black screen and no LED on front).
Booting up on either machine did not work *until* some time passed (overnight in my case) with the Cinema HD disconnected and unplugged.
In decades of using the Mac, I have never seen a pram reset solve anything but does mess up alot of things 99.9% of users would rather leave alone and unmolested. Techies love PRAM reset as a first go to because it is a lazy quick fix. But, it actually never fixes most problems. It is overkill 99.9% of the time.
SMC reset for Intel Macs is just unplugging the computer for 15 seconds. Again, have never seen this fix any Mac problem in decades.
I won't debate with you except to point out the number of times smc reset has helped users in these forums and to a lesser extent pram reset. And maybe it was your experience in older machines but these do not really change much of anything significant to the user but may clear some ills up. They are particularly useful for many bluetooth problems.
Speaking for myself I use them whenever I make a configuration change (add/remove ram, connect/disconnect a monitor, add/remove an internal drive) in my mac pro. I know it works in those cases because almost every time I change configurations I cannot sleep my machine. Attempting to do so causes the fans to rev up and the machine to immediately wake up. And 100% of the time that this occurs the smc/pram resets fix the problem.
Fortunately changing confiurations does not happen that often for me. But when it does smc/pram reset is an easy fix. And nothing else suffers as a result of it.