Re: scientific names, very true.
Common names are, at best, descriptive, but are difficult to relate to what the requirements for keeping said critter are.
As I understand it - and bear w/ me, I'm a coral amateur - there are several different distinctions as I understand them.
Soft corals = no calcerous skeleton. Among them, Zoanthids/Palys, Mushrooms/Ricordea and Polyps of various kinds (stars, yellow, etc.) Related to anemones, same family, but not the same genus. Largely photosynthetic, but not always so. Those which are photo are easier to care for. Some require supplemental feeding, some don't.
Hard corals have calcerous skeletons, and are divided into two types:
Large polyps and small polyps. (Frogspawn, etc. fall into the Large Polyp category)
Both are, to an extent, photosynthetic, but their growth is enhanced by salt composition, feeding and specific lighting - both in intensity (lumens - or very generally, watts per gallon) and light frequency (expressed either in angstroms or °Kelvin).
Small polyps (SPS) have large skeletons compared to polyp size - branching corals such as Birdsnest, Acropora, etc., whereas Large Polyps have small skeletons comparitively - Frogspawn, Hammer, Torch, etc.
SPS require greater light intensity than LPS, hence the debates over T5, Halide, PC bulbs. - I.e. SPS require the intensity of Halides or at least T5's, which LPS require less of. Soft corals are pretty much ok with PC's which can provide the frequency, but not the intensity of the former two.
All these things are depth and volume dependent as well - i.e. a 20g Long has more photosynthetic capability than a 20g High, per watt.
This is my general understanding of the divisions at this point.
Those who understand better - please chime in - and do please correct me if I'm wrong. I'm trying to learn here too.