According to Oxford Dictionaries Online:
Predicator means "(In systemic grammar) a verb phrase considered as a constituent of clause structure, along with subject, object, and adjunct."
Predicate means "The part of a sentence or clause containing a verb and stating something about the subject (e.g. went home in John went home)."
In the latter example of John went home, it seems, the predicate is went home whereas the predicator is went, home being an adjunct and thus not part of the predicator as defined.
Now, I looked further in this wikipedia article to better understand the difference. The article recognizes two competing notions of the predicate in theories of grammar:
(1) Predicates in traditional grammar (e.g., went home in John went home)
(2) Predicates in modern theories of syntax and grammar (e.g., went in John went home)
Which, according to the article, causes confusion as to what exactly the term predicate mean, and some grammarians came up with a new term "predicator" specifically for use (2), says the article. No problem thus far.
What bothers me: The article in its explanation of (2) says, "Other function words -- e.g. auxiliary verbs, certain prepositions, phrasal particles, etc. -- are viewed as part of the predicate." (Emphasis mine.)
Now, remember this definition of the predicate, i.e., (2) above, corresponds to the new term "predicator".
I understand that auxiliary verbs are part of this definition of predicate, because an auxiliary verb can be part of a verb cluster. But so can certain prepositions and phrasal particles?
The article has these example sentences (Words belonging to "predicate" (2) are boldfaced as in the article itself; My comments in parentheses.):
The butter is in the drawer. (preposition in being part of the predicate)
You should give it up. (particle up being part of the predicate)
Susan is pulling your leg. (I don't know why leg is marked as part of the predicate. Maybe a typo?)
Except for the last one, which I suspect is a typo, I'd like to know whether the preposition in and the particle up belong to the predicate as presented in (2) and thus belong to the new term "predicator".