Probably this is question is too big to make a proper response. But I'll start with some ideas.
Color temperature has different meanings in design (and related arts and science)
1) Psychological aspect
This is how we perceive color. We perceive red as a hot, because if we put a needle on the stove it turns red and we know it is hot.
We perceive orange and yellow as summer time and white and blue as cold. This relationships are how we perceive color in daily basis.
In a chromatic circle we separate colors in hot and cold... of course we need to define something in between, so the ones that are not red or blue (green, purples or magentas) we just call them warm... But probably that has not much sense.
A logo could use this basic relations. A red logo energetic. A blue one calm.
2) Color temperature in physics and photography
As you probably know color temperature is a theoretical model of heating a black object, so the more intense heat generates more energetic electromagnetic waves.
This is how we know the temperature of a star for example.
But this turns out that the color wavelength is backwards as we perceive them normally. A red color is cooler than a blue one. White is in the middle... Actually white is just a perception of a combined range of color temperatures.
Color temperature is a crucial factor in photography. It is not about the real temperature, but a way to balance the white point.
It is funny how we perceive a reddish photo as a warm one and a blue or white one cold.
Coming back to your question
that color temperature, brightness, saturation...
Color temperature is an aspect of Hue, the chromatic nature.
Normally color models have a 2 dimensional base and a 3d component for variations.
Color temperature is one linear dimension that is complemented with tint, and thoose 2 are very limited. You have some mixed ideas on the color models. Color temperature is not necessary the same as hue, or HSL or HSV models.
for me color is evaluated just by the RGB values allowed in a browser. Most of my interest lies in finding more formal ways to think, track, balance and document color choices.
RGB values allowed in current hexadecimal technologies are just the options you have.
But the first design aspect is How do you make choices. That is a very complex question.
Balance is a "fine tuning" point of the design process. Track probably is an administrative aspect (given in a design manual).
And think color... I hope that is what makes a designer a designer. :0)
Edited some time later
You do not think on any project as one color. You could not see any black text on a black background. You need at least two colors.
This implies that one aspect of thinking in color are color relationships. For example, you could have a ,saturated subject on a non saturated background, a warm object on a cold background, or more specific, a warm main light with a blue ambient light.
This kind of topics can be found using a more detailed search on google, like
Or even more specific on adjacents fields of design.
Color is not an isolated topic. It is in every day life. So it has psicological, practical, technical, cultural compenents, etc.