I am an electrician at a commercial electrical company. There is an in-house training program that posits the datum that when talking about the designations of color temperature for lights (for example an LED light with a 5000 Kelvin color temperature) that the designation means that the light is that temperature, thermodynamically....but that would mean that the light is 8540.33 degrees Fahrenheit. Yet LED lights are not only known for running much cooler than incandescent, but I mean, the idea that the light has over 8,000 degrees of heat seems illogical. My position on it is that when using K when talking about color temperature is that we are strictly talking about color, exclusive of heat or lack thereof. I am not sure and need help sorting this out.
Your intuition is correct. For a normal filament type bulb we run current through a narrow wire and the current heats it up due to ohmic resistance and it achieves something like thermal equilibrium - and the physical temperature and the color temperature are the same. This means that there is a lot of infra-red which is perceived as being hot. This is a real waste of power unless you are using the bulb in an "easy bake oven".
With the LED bulbs the LED's emit only a few wavelenghts of light and are definitely not in thermal equilibrium. Semiconductor junctions are used to emit just certain visible frequencies. Then, usually, the LED's are surrounded by phosphorescent material that absorbs the LED light and "smooths it out" by re-emitting over a range of visible frequencies. When assigning a "color" to these LED bulbs the visible section of a thermal radiation curve (a.k.a. black body radiation) is compared to the light of the LED and a color temperature is assigned. However, this assignment is partially art and partially science (and largely regulation) since the LED's are not thermal emitters.