As they learn more about the chemistry of red giant stars' atmospheres, it becomes obvious that the chemical evolution of one may be very different than another.
Spectral classes are not only determined by temperature, but also by their chemistry.
The chemistry indicates age, different origin conditions, different constitutions of elements, probable future evolution and fate and possible variability.
Once the mass and chemistry is known, it can also lead to a determination of distance and the amount of material lying between us and the star.
If the distance is known exactly, from parallax, then the probable size and surface temperature can be calibrated against our knowledge of the chemistry.
Red stars were once in classes R & N, but are now split into several sub classes of C based on all those factors.
If you have not followed this for a few years, just know that the old classes of OBAFGKM have all been further divided into sub classes and new classes
as our knowledge of stellar evolution and chemistry has evolved.
One thing to be aware of as an observer is that carbon stars tend to get much redder as their atmospheres expand (a lowering of surface temperature)
and get more orange as their atmospheres shrink (a raising of surface temperature). Carbon stars are much more dramatic, color-wise, on the down cycle of their variability.
As one example: in October, Omicron Ceti (Mira) was at its brightest. It's now on the way down.
I noticed it was a yellow-orange when I observed it late last Fall. I caught it near minimum a few years ago and it was quite red.
The visible light shifted into the infrared, leaving mostly the deep red visible. It was cool to not only see the range of variability, but also the color change.
V Hydrae, visible now, is like that as well--on its way down, it gets redder and redder.