CRI is absolutely real. It's an old and relatively simplistic metric with several potential successors. The chart you linked uses one: TM30, which is based on the average of 99 different colors instead of CRI's 8.
There are seven extended samples for CRI (R9-R15) not included in the average. LEDs often do particularly poorly on R9, a measure of saturated red rendering. LED sources with high R9 usually advertise it separately.
Tint, or blackbody deviation (Duv) is also important to the look of light and listed on the chart, but not for every model. These numbers are very small, but important: anything outside of +/s 0.006 is not white light according to ANSI. +0.006 looks very green, and -0.006 looks very pink. Interestingly, after acclimating for a few minutes, most people think very pink looks better and more natural than neutral white[0]. Most people do not like green tint.
There are seven extended samples for CRI (R9-R15) not included in the average. LEDs often do particularly poorly on R9, a measure of saturated red rendering. LED sources with high R9 usually advertise it separately.
Tint, or blackbody deviation (Duv) is also important to the look of light and listed on the chart, but not for every model. These numbers are very small, but important: anything outside of +/s 0.006 is not white light according to ANSI. +0.006 looks very green, and -0.006 looks very pink. Interestingly, after acclimating for a few minutes, most people think very pink looks better and more natural than neutral white[0]. Most people do not like green tint.
[0] https://www.energystar.gov/sites/default/files/asset/documen...