First of all Kelvin temperature scale is the color of a black (no reflective light from other source) object at certain temperature. Hot object emitted electro-magnetic radiation (radio waves on one end to gamma radiation in the other end with everything else include visible light in the middle). Regarding the visible light spectrum, the hotter they are the bluer they look. Slightly hot object look reddish. The sun spectrum is 5500K (surface temp of the Sun) but true full spectrum (all wave length of the visible light are there). Bluer stars have hotter surface temp and other stars like Red giant have cooler surface.
Indicated label spectrum of our bulbs are not full spectrum but just apparent color to the eye. There is not a standard what color should be call what. 10000K from one brand may look like 6000K, or 14000K of another. They can call it whatever they want.
Zooxanthallae photosynthesize and depends on zooxanthallae species, the peak spectrum that is use is different for different species. Those that in deep corals and anemones species use bluer wavelength while those on the surface use more of the other end of the spectrum because blue light penetrate the sea water better. Things get bluer and the dept increases. Corals and anemones have different and multiple zooxanthallae species in their tissue and the population will change with condition.
That is why they adapted to different light conditions.
With all that said, Most people use (or should use) bluer light more for look than for photosynthesis. IMO, I would use about 10,000K. I think it is better for the corals and anemones than the blue Radium.