Let’s call any sentence in which a modal word such as “possible”, “necessary”, “impossible”, “can”, “cannot” etc. plays a decisive role in determining the truth of that sentence a “modal sentence”.
A modal realist(MR) would subscribe to the following:
a) There are modal sentences.
b) The truth values of some modal sentences are fixed by the behavior of something(s) that is not both actual and concrete. (call these things m-entities)
c) M-entities exist.
A genuine modal realist (GMR) would add the following:
d) Some m-entities are concrete
An abstract modal realist (AMR) would add the following:
e) No M-entities are concrete.
Note that (d) -> (f)
(f) Some m-entities are non-actual
And that (e) is compatible with (g)
(g) All m-entities are actual
Note that (d)-> ~(e) and (e) -> ~(d), thus GM and AM are exclusive. Note also that ~(d) -> (e) and ~(e) -> (d), thus GM and AM are exhaustive.
A couple things are worth noting here. If we're speaking of possible worlds (for example), the m-entities would be all merely possible worlds. I'm not trying to parse out what different theories think of the actual concrete world (and actual concrete objects), since that is trivial (I hope) and would make the characterization needlessly confusing.
It may seem that (b) makes (c) redundant, but I threw (c) in to exclude noneism.
The versions of abstract realism he mentions in the book still fall under AMR under my characterization. In addition, Soames would count as an AMR under my characterization.
A fault of this characterization is the extremely vague notion of a "modal sentence" I employed. I'm still kind of fuzzy on the whole relationship between everyday english usage and the more strict philosophical talk. If only english had boxes and daimonds!