(Been meaning to post a more detailed response and was reminded by another thread I saw. :/ )
To start with the video in the OP seems to strongly infer (starting about 2:10) that in a normal healthy corals the bright colors are made by the zooxanthellae with the coral only making color at the ends. To be clear, it should be pointed out zooxanthellae only provide brown coloration to the coral. All the fluorescing and chromo proteins that provide color are being made by the coral.
Research clearly shows corals make flourescing and chromo proteins to deal with less than ideal conditions. When we manipulate environmental conditions to force them express more color we are creating less than ideal conditions for them. To what extent that's bad is debatable. But since we can't quantify a healthy coral by coloration it seems to me we need to be looking at what research is showing to be best for corals, not what brings out a desired color. Innate immunity of a species/genotype/variety is a significant factor in a corals survivability under stressful conditions. While one coral may survive a set of condiitons another coral with a weaker immune system will do poorly or die. I would add while the only coloration zooxanthellae provide is brown, meleanin is an important part of coral immune systems and may color corals brown as well. For those interested here's links to some of the research:
Optical feedback loop in colorful coral bleaching (I know I already posted this video but I like it).
Just to point out white may not necessarily be bad.
White but not bleached, healthy corals without bight colors
Climate change and other types of environmental stress are known to increase corals’ vulnerability to bleaching, a process whereby colonies lose their colour either due to the loss of photosynthetic symbionts or their pigments. Although bleaching
www.academia.edu
Coral level of immunity variability A coral's inherent immunity effects how it respones to stress
Disease causes significant coral mortality worldwide; however, factors responsible for intraspecific variation in disease resistance remain unclear. We exposed fragments of eight Acropora millepora colonies (genotypes) to putatively pathogenic bacteria (Vibrio spp.). Genotypes varied from zero...
www.nature.com
Roles of coloring proteins
Photoprotection
researchonline.jcu.edu.au
High light levels induce bleaching and is additive with high temperature
PDF | Elevated temperature has been implicated as one of the main causes of coral bleaching (= dissociation of coral/algal symbiosis). However, there is... | Find, read and cite all the research you need on ResearchGate
www.researchgate.net
Fluorescing and chromo proteins used as antioxidents to neutralize excess oxygen from photosynthesis
Background A wide array of fluorescent proteins (FP) is present in anthozoans, although their biochemical characteristics and function in host tissue remain to be determined. Upregulation of FP's frequently occurs in injured or compromised coral tissue, suggesting a potential role of coral FPs...
journals.plos.org
Flourescing protein used to deal with internal parasites.
Caroline V. Palmer, Melissa S. Roth, Ruth D. Gates, Red Fluorescent Protein Responsible for Pigmentation in Trematode-Infected Porites compressa Tissues, Biological Bulletin, Vol. 216, No. 1 (Feb., 2009), pp. 68-74
www.jstor.org
Lighting spectrum influence on coral, differential responses by corals to light
Departments of Chemistry and Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802
reefs.com
The morphological plasticity of scleractinian corals can be influenced by numerous factors in their natural environment. However, it is difficult to identify in situ the relative influence of a single biotic or abiotic factor, due to potential ...
www.ncbi.nlm.nih.gov
Melanin
One concern are we selecting for genotypes/varieties that have reduced melanin production and are inherently more sensitive to bacterial infections and bleaching.
Disease epizootics are increasing with climatic shifts, yet within each system only a subset of species are identified as the most vulnerable. Understanding ecological immunology patterns as well as environmental influences on immune defenses will provide insight into the persistence of a...
pubmed.ncbi.nlm.nih.gov
Nutrients
"When I see the colors of some of these low nutrient tanks, I can't help but be reminded of bleached coral reefs. It should therefore not come as a surprise that feeding corals in such systems becomes a very important component in these systems. Though reefs are often catagorized as nutrient "deserts" the influx of nutrients in the form of particulates and plankton is quite high when the total volume of water passing over a reef is taken into consideration.
Our crystal-clear aquaria do not come close to the nutrient loads that swirl around natural reefs. And so when we create low-nutrient water conditions, we still have to deal with the rest of a much more complex puzzle. Much like those who run their aquarium water temperature close to the thermal maximums of corals walk a narrow tight rope, I can't help but think that low-nutrient aquariums may be headed down a similar path." Charles Delbeck, Coral Nov/Dec 2010, pg 127
Southhampton did a series of expereiments with various corals maintained in a closed system for over 8 years. Here's an image from one of their paper "Phosphate deficiency promotes coral bleaching" showing the effects of different Nitrate

hosphate ratios on a Euphylia Sp. (keep in mind corals are also using particulate and organic forms) . Ideally it seems what would help us tremendously would be comparison shots of every coral commonly kept under different conditions, hopefully someday. Listed below are some of their papers.
An experimental mesocosm for long-term studies of reef corals - Volume 92 Issue 4
www.cambridge.org
An Experimental Mesocosm for Longterm Studies of Reef Corals
Phosphate Deficiency:
Nutrient enrichment can increase the susceptibility of reef corals to bleaching:
Increased dissolved inorganic nitrogen (DIN) concentrations in sea water have been linked to a reduction of the temperature threshold at which corals bleach, however, the mechanism underlying this change is not known. This phenomenon is now explained in terms of increased phosphatase activities...
www.nature.com
Ultrastructural Biomarkers in Symbiotic Algae Reflect the Availability of Dissolved Inorganic Nutrients and Particulate Food to the Reef Coral Holobiont:
Reef building corals associated with symbiotic algae (zooxanthellae) can access environmental nutrients from different sources, most significantly via the up...
www.frontiersin.org
Phosphate deficiency promotes coral bleaching and is reflected by the ultrastructure of symbiotic dinoflagellates
Enrichment of reef environments with dissolved inorganic nutrients is considered a major threat to the survival of corals living in symbiosis with din…
www.sciencedirect.com
And for those carbon dosing or nitrate dosing to accentuate colors:
onlinelibrary.wiley.com