Correct, LPS and SPS have no relationship with the scientific classification of corals.
The funny thing is, based on DNA evidence, there *are* two broad categories of corals. Robust and Complex. These coral lineages have been separated for a very very long time over evolutionary history (>400 my) and are different in many ways. If hobbyists are looking for a general two-level classification of corals we should use these, instead we use LPS and SPS which appear to have little or no overlap with the phylogenetic classification.
Personally, I think we're probably best served by talking about these things at the Family level , since this seems the most useful in terms of the corals' needs. For example:
Pocilloporidae - includes Pocillopora, Stylophora, and Seriatopora
Acroporidae - includes Acropora, Montipora, Anacropora
Dendrophyllidae - includes Tubastrea, Dendrophyllia, Balanophyllia, Duncanopsammia
of course LPS is a lot easier to write than Dendrophyllidae but the point is that two different "LPS" corals (e.g. Duncanopsammia and Platygyra) may be more distantly related to each other than a mammal is to a bird... while each is much more closely related to other corals we call "SPS".