A lot of good points, I'll just add that it depends on the coral. Some corals do best in clean, environments, with very low nutrient levels. Others will do better with slightly elevated nutrients. It's sort of a generalization, but I think corals with a higher tissue to skeleton ratio (LPS and soft corals, which don't have a skeleton at all) do better with slightly higher nutrients. Corals, such as SPS often will do better with very low nutrients. Also, another generalization, but many SPS can change their pigmentation to adapt to lighting and may require more light to begin with. Some LPS, for example, may not be able to adapt their pigmentation to different lighting and may not require as much to begin with depending on the clade of zooxanthellae hosted for example, and other factors. So, matching lighting to the particular coral is important. Same for flow, some corals, such as LPS can expand their shape, to allow more surface area for gas exchange, but may be damaged by too much flow. Others, can't regulate their shape and require lots of flow, for access to minerals and nutrients and for gas exchange.
So, in short, the most important factor for growth is to provide an environment that is well suited to the corals being grown.