Correct answer is all of the above, but in varying degrees. Which one actually makes a discernible difference with your fish will depend on the species and the sum of the other factors. Color in some fish is more water quality sensitive-- or in some species can even be sensitive to which minerals are present in the water and in what proportions-- some fish are more nutrient sensitive, some more variable according to both tank and ambient lighting conditions, some are very changeable according to mood or can vary according to status within a group, spawning status, etc. That's one reason you'll get different opinions, it depends somewhat on which fish you're talking about.
Another reason you'll get different opinions is it can also be situation of
everything else being equal, this or that makes the difference. Example: everything else being equal, some species are more sensitive to diet. Or, everything else being equal, water quality is the primary variable or everything else being equal, background and lighting. I've done a LOT of food experimenting, some species look better on one while another species will look better on another food or another species no real difference either way-- part of the reason for arguments on this imo. I bred red head geos for several years and I can make them more orange or more red according to which food I give them.
Lighting is it's own whole subject, in terms of spectrum of your particular bulbs. Differences between bulbs can wash out, intensify, highlight, distort, muddy, or clarify not only fish color but also plants, rocks, etc.
The "enhancement" is all visual on the dry side of the glass. Certain colors will be visually more appealing because of color and contrast. I think you are saying the same thing with the shirt example. My point was that the fish won't actually change color because of a background or substrate. That has to be a more direct physical reaction to diet, genetics and water.
That can be true up to a point, but apparently you've never kept Cyphotilapia, especially gibberosa, which
absolutely change according to environmental color and lighting conditions and can darken or lighten themselves dramatically and can also change hue-- more blue, more purple, more black, etc. So that may be true of some species, but not all. Cyphotilapia are also very mood driven or can look very different in one tank vs. another based purely on lighting, background, substrate, what other fish are in the tank, etc. Again, it can depend on species.