My Xerox Phaser 6128MFP printer flashes a red light with a "Load paper in Tray" message, when the specific paper size and type are already same in the tray.

If the paper size is correct in the printer and the printer and it's possible to print off a test print in both the printer control screen or directly from the print driver utilities if that works then the only thing I can suggest is the item you want to print is defaulted to a different size. i.e. Check that the word processor your using is set to the size you have in the printer. Imperial size pages will look very similar on screen as does the A4. Printers don't automatically fix paper print sizes you just have to check and recheck for differences. Removing the tray only reafirms to the printer that the size might be incorrect and your going to fit the correct size.
As for the red light all printers behave differently in certain error conditions. Have you also tried to print out something you printed at a time the printer was working correctly. In other words try printing out a different file. Maybe, trying to print off a test print of one line of the alphabet, and then go from there.
Good luck,