We usually project using a HP laptop which results in the same overblown effect.
Yes, it’s interesting that the same effect happens on both the MacBook Pro and the HP laptop. That would suggest it isn’t specific to the Preview app on the Mac, and it’s probably not specific to macOS or Windows.
And it might also mean it isn’t specific to calibration, if it does the same thing even though the Mac and the HP probably apply different default profiles for the projector. Although I do recommend calibration, I wouldn’t expect it to fix that big difference between normal and full screen mode. Because a profile should have the same effect on both.
What I am wondering if there is something about the projector that processes images differently for full screen mode (regardless of Mac or Windows), like some kind of automatic in-projector image quality “correction” or “enhancement” that gets applied in only one of the two modes. For example, suppose the projector assumes full levels (0–255) when showing a normal computer desktop, but assumes
video levels (16-235) in full screen mode because it thinks you’re now watching a movie…maybe that could produce unwanted clipping?
But I don’t know very much about projectors, you’d have to look in the projector manual to see if that is even a thing. If the third party is unable to provide the manual, look at the model number and see if you can find a PDF version of the manual online.
If you do find the manual, another thing to check is whether the projector has different display modes. Maybe it’s on some kind of Auto mode that tries to “enhance” everything coming through, and maybe there is a button on the remote that can switch it to some kind of un-altered “native” or “passthrough” mode that would be more suitable for critical photography viewing.
If you did have the opportunity to calibrate it, I would certainly investigate which mode the projector is in during calibration. Just like TVs and computer displays, if it’s in a mode that does too much dynamic correction/enhancement all the time, it will change its behavior so frequently that a profile won’t have a fixed baseline to correct.