As "white" is really the presence of all colors, …
That's not quite correct: there is no such thing as `white' unless you specify a color temperature. Essentially what we perceive as `the most natural white' are the emissions of a black body radiation source such as the sun (even then, the atmosphere absorbs different wave lengths at different rates so that the sun light we see does not follow Planck's black body radiation formula). The sun's surface temperature of 5,500~6000 Kelvin is also the most natural white to us there is.
Any light source with a different spectrum, e. g. CRTs, CCFL-backlit displays and LED-backlit displays, that's why the different types of monitors and backlights will display colors slightly differently even after calibration. (Have a look
here for a comparison of spectra of different light sources). For CRTs, for instance, blue and green pixels look relatively `smooth' whereas the pixel supplying the reds contains many small peaks at higher wave lengths:
Compare that to a white LED:
and LEDs can only emit in specified spectra, there will never be a true "white" LED.
White LEDs are not different from CCFL tubes in that respect: fluorescent light bulbs use an electric current in a gas (e. g. mercury) to produce UV radiation. There are a few clear peaks
in that spectrum that are due to mercury's particular electronic structure. Then a second layer on the surface of the glass converts UV radiation into visible light.
This principle is also used in some types white LEDs: they are blue LEDs with a coating which converts some of the blue radiation to larger wave lengths. The other type of `white' LEDs consists of a red, blue and green LED where you can mix the color temperature at will.
To be honest, I'm not sure which type of white LEDs is used in backlights (perhaps both?).
Currently I thought that the "white" LEDs were simply a red, green, and blue LED combined into one unit.
There are several different types, the one you describe being the most sophisticated, because it can actually change the color temperature (if the electronics of the monitor support that).
The other type is the one described above which is a blue LED with a coating that converts some of the short wave length light to larger wave length light.
The end result is that monitors with different types of technology render colors differently. To my knowledge, LED-backlit monitors tend to have a harder time differentiating reds.