What are you using to change your "white reference level"?
Normal controls are labeled "brightness" and "contrast/picture". "Brightness" will adjust the level of the darkest black, and "contrast" will adjust the level of the brightest white.
What you describe, "setting white to reference level makes picture look washed out". Normally raising the brightness/black level too high makes the picture look "washed out" (no deep darks), and lowering the brightness too low will lose shadow detail. Raising contrast/picture level too high will lose detail in bright areas (and possibly cause banding), and lowering it too low makes the picture look "dull" (no bright areas).
Make sure you are adjusting the correct user control.
BTW, I don't think the scene in The Matrix you mention is supposed to be pure white. If you were to grab a frame from that scene and see what "color" the white is, I'm sure you'd find it is a light grey. So if you try to use that scene to set your white level (contrast/picture setting), it will make actual high whites too bright, and you will lose detail.
I'm not sure what your projector has for gamma setting options, but you want the standard 2.2-2.5. You'll have to find out what the proper gamma setting for your projector is, assuming it has that adjustment available.
If you don't know what the gains/cuts do, don't touch them. You could really mess up your picture. Basically they are for adjusting your greyscale color temperature. If you think your greyscale is too warm/cool, try the color temp settings (Warm/Cool) in the opposite direction first. Otherwise you could really mess up how your picture looks.
Go back and redo your brightness/contrast settings correctly, and don't worry about the Matrix scene. If you have the Finding Nemo DVD, you might even try the scene I mention in my original post, and see if anything happens if you turn the contrast/white level up too high (extreme banding, etc.).
Your goal should be the proper black (16) and white (235 or above) settings with smooth gradations throughout (barely visible steps between 16-black, steps from 17 on up, and the white steps at the top end), correct gamma curve (whatever setting that is for your projector), proper color temperature (will appear slightly yellowish, at 6500K), proper color saturation and color balance. Make sure you have all the factory defaults written down, and don't mess with things you don't know the function of unless you can restore them to the default setting.
It may take some time, but you'll start to get it. I know that when I started this thread, I myself didn't have a 100% firm grasp on it, especially compared to what I know now (after 1 year of tweaking my service menu, messing with my HTPC, etc.). You might also see if there is an official thread on your particular projector and see how people have tweaked it.