On the Electowiki proportional representation page there is an explanation about the different interpretations here. A new philosophy called Proportionality for Solid Coalitions has recently been added to this list. To my reading this seems very similar if not identical to Monroe’s method.
I added it, so I’ll explain it. Basically, Monroe takes into account utility, whereas PSC doesn’t. 1-winner example:
B has the most utility within their most-supporting Droop quota, so Droop-based Monroe would say that B is the best candidate. But Droop-PSC would say that A is best (in the single-winner case, it is the mutual majority criterion).
I thought Monroe’s theory was originally Ordinal based.
In the single-winner case, FPR with ordinal balloting reduces to the Borda count.
Basically, PSC focuses on sets of candidates being preferred above all others by quotas of voters, whereas Monroe is totally about utility.