Color fidelity issue when exporting - colors changed

What you're expecting is near impossible without expensive calibration gear, and realistically, not necessarily that much better even then.

Why? I work for/with/teach pro colorists, who deal with this all the time. I've got years of experience with trying to match screens. And endless hours talking this through with some of the best teaching colorists out there.

You can take two "identical" monitors, connected even via a BlackMagic or AJA external device, so that the computer OS and GPU aren't 'touching' the signal at all. Calibrate them with say a Klein spectro unit (around $5,000 or better) .... create a color calibration LUT for each ... and then run the same signal to them sitting side by side.

Most people will see at least a small difference between the two screens ... ! That's how hard it is matching images between screens, within an Ideal professional setup and calibration routine.

Watching something on the same screen, say a tablet, on a park bench at noon, and a dark bedroom at night, will have a different look. Browsers, monitors, players, TVs, all will have a different 'look' to them.

So, lets get started with your situation. I don't know how much of this you know, and there'll be a lot of inexperienced folks reading this over time anyway, so I'll assume not a lot. Don't take offense if you do know most of this ... 

Have you done any calibration on your monitor or projector, with a puck and software ... like the Xrite i1 Pro unit and software? If not, I can guarantee you can't be sure either your monitor or that projector is even close to "accurate". Which is a total bummer, but that's Real Life in Video Post Production.

I've had monitors designed for video post that come with fancy certificates of how tightly they passed a Rec.709 calibration test. And ... on testing on my setup, they were WAY over 100 nits brightness, and the color performance was skewed well off the sRGB range.

So they were not suitable for my needs at all until I ran a calibration with the i1 Pro, then ran a profile pass ... using the ColourSpace calibration software, using Resolve as a "patch generator" ... and checked what the monitor was actually putting out after the calibration. I don't use it until I get a profile with the expected tight adherence to Rec.709, and particularly that delta-E reading.

Back to your monitor and projector. Another question. Have you checked to remove all the settings that monitor and projector makers throw onto the signal "to enhance the viewing experience" ...? Such things as juicing the contrast or saturation, or setting it a bit "warmer" or too bright to properly work? Nearly all screens come with crud for settings.

And then ... are you working in SDR ... Rec.709, or say the HLG form of HDR? We would need to know to make sure your color management settings are the best they can be in Premiere.