When I print my photos, they usually have lots of blue sky. When I view them under real daylight, it looks normal, as if it was real sky.
When I view it under 3000K LED light, the blue gets more saturated, as if the 3000K is a lie. It also looks that way when I view the photo under 6500K triphosphor fluorescent light.
The question is, does the 3000K color temperature mean essentially nothing when viewing colors?

Pretty much so. You hit the limitations of light sources coming from what they got in their spectrum, which some peeps cant see in the general appearance of the light

My opinion is that HPS has copious amounts of blue light, it's just that the orange overpowers the blue. When examining the HPS spectrum, that's clearly visible to me.

So what does C.C.T. and C.R.I. really indicate? How are these defined? Why do we need them and still use them? Aren't they relative to something? I have no concrete answers to these questions myself. If you can answer any of these even superficially, you might glimpse a little of the truth.

I think CCT is the color of the light when the light source is lighting up a white object.
I think CRI is how colors are shown correctly.

So what does C.C.T. and C.R.I. really indicate? How are these defined? Why do we need them and still use them? Aren't they relative to something? I have no concrete answers to these questions myself. If you can answer any of these even superficially, you might glimpse a little of the truth.

CCT represents a temperature of a black body radiator, where the correlation of the emitted spectra is the highest between the black body and the tested light.

CRI means a measure, how well are things of different colors rendered, again compared to an illumination with a reference black body with the same CCT. It is evaluated on a set of test color stripes (their reflection spectrum is standardized; practically all are wider band filters), each test color evaluated separately and then the CRI is just an average of those individual results.
The thing is, this is by far not covering everything, the methodology is designed to rendering of rather natural objects, the color printed photo is not a natural object.

The photo printing uses certain color selective dyes, which are then mixed on the paper. The thing is, the reflected spectrum is practically not that much specified, just the whole printing process (include software calibration and color transformations) is caliobrated to yield good displayed colors for an ideal illumination (usually natural daylight).
The thing is, the reflection spectrum of the dyes is not wideband, but narrow band peaks (if they want to display really extremely wide color range with just 4 dyes, they have not much other options). And there is the problem: The result is dictated, how the reflection spectrum of the printed picture matches with the spectrum of the light source. If the dyes are wide band ones, the color accuracy remains good even with peaky style light source spectrum (most artificial high efficacy light sources), when the dyes use narrow band reflections, the color reproduction becomes bad under those peaky light spectrum.
The reason for using narrow band dyes is, how wide color range they want to cover. With wide band dyes you can not display saturated colors, the narrow band dyes can go much further.
Usually with cheap printers, where the contrast of the printing head is limited (they want to use more water, to save money on dye chemicals), the color range becomes limited further. And top compensate that, the cheap printers tend to use the narrow band dyes - with that they are able to reach somewhat usable color range in a cheaper way. The cost to pay is, the color becomes way more distorted, when you use narrow and spectrum light instead of the broadband daylight and/or incandescent.
The good printer makers do test and optimize their dyes not only against the ideal lighting conditions, but as well with the common light sources. This testing is, what makes a big part of the toner cost, so the cheap vendors skip such optimization, so their toners are then way worse adopted towards the nonideal illumination.

What about pigment-based printers, or printers with many more ink selections?

What about pigment-based printers,

Well, I maybe do not understand this question:
As far as I know, except for some special cases (engravers, hologram printers,...) all printers always use pigments (so dyes) to form the desired colors. So the description above was written for the dye based printers.
The hologram ones are usually either laser or electron engravers forming an interference pattern, which then leads to the desired color (and eventually 3D effect).
And engraver just forms tiny holes, which then form the desired pattern (usually used to label parts without using any extra material at all)

or printers with many more ink selections?

Using more than the basic "CMYK" is indeed the way, what many high color quality (mainly professional) photo printers use. But it means many more than just 4 color cartridges, so we are talking about quite expensive piece of equipment...
In fact the presence of black toner is already a small step in that direction - all colors could be theoretically formed by just the 3 "color" toners (CMY), the black is there to enhance the BW contrast (mainly for texts,...) of the image without the need for too thick basic color toners...

Most inkjet printers I've seen don't uses the black ink for the photos, but only for the text, and hence the lower cost of the black ink compared to the CMYK inks. My Canon IP5300, have two black inks, one PGBK for text and one BK for the CMYK inks for photo printing. The BK black cartridge is more expensive than the PGBK black cartridge.

Well, I maybe do not understand this question:
As far as I know, except for some special cases (engravers, hologram printers,...) all printers always use pigments (so dyes) to form the desired colors. So the description above was written for the dye based printers.

Dye-based printers use dyes for their color, and that yields bright colors, but they fade quickly.
Pigment-based printers use pigments for their color, and they can last a very long time without fading, but the colors aren't as bright.

The pigment is the main chemical substance (usually a fine powder, even in the liquid inks) giving the final color to the dye.
The dye is the mixture, where the pigment is dilluted into some auxiliary material allowing the further mixing of the ink/paint (the powder pigment mixed within some water with tenside or so)...