There's only such thing as real color temperature with incandescent lamps.
For discharge lamps, it is called correlated color which is the closest match to black body radiator color based on the chromaticity (expressed in x and y coordinates) of the lamp. It's like regression analysis.
you can have one light that's greenish and one that's pinkish and still have the same CCT "K".
CRI is more precisely called Ra8 color rendition and its an arbitrary measure. It is based on measured rendition of eight CIE standard color chips. It's an outdated system in my opinion.
It means Rendition average of 8 chips.
There's another arbitrary measure called Ra14 which is computed the same way but six additional color chips are used.
Many trichromatic lamps are designed to look good on Ra8 test, but almost all of them have a R9 rendition of close to zero which is the deep red color chip used in Ra14 test but not in Ra8 test.
So, a GE SPX30/Philips TL80/Sylvania830, etc are all 3,000K CCT with ~85 CRI, but has a R9 of about 0, while a 3,000K halogen has an R9 of 100, and Ra14 and Ra8 of about 100.
There's pleasing distortion (GE Reveal, meat display lamp), visually accurate (Colortone 50 etc), spectrally accurate (film or scientific use, like xenon high pressure of filitered 5000K incandescent).. what's good depends on application.