The black and white clipping patterns are created the same, so if one works then the only clear guess on why the other wouldn't work is if you're clipping levels somewhere. If you are using the current patterns, you definitely should be able to see flashing on the white clipping pattern. The only CRT I've looked at clearly showed a few levels over 235, so in general yes you want to see flashing with the pattern.
The first thing I would do for testing is to take the receiver out if it's in the video chain. Just hook the video source directly to the projector. For simplicity my general recommendation is to try to output from the source at typical video levels, don't alter the signal with other devices, and then to do adjustments at the display. Generally that means for most sources you would probably want to leave any adjustment controls they might have at the default settings, but it is possible that the source could be clipping so you might have to test there also. Many video devices have their own adjustment controls, so the recommendation to take out any receiver is to simplify things. With just a source and a projector you only have two places where the signal could be clipped, instead of three.
There are further items around this issue that could be brought up, but first I would test with just a source and a display. I would set the source to default settings if it has video adjustments, and then I would turn down contrast at the display to look for flashing. If there's no flashing with reduced contrast at the display, only then would I turn down contrast at the source. If after doing this the white clipping pattern still doesn't flash, I'll mention the other possibilities.