Okay, so after a bit of thought, here's a guess (one that will likely make it into my thesis now...):
Bybee's Resultative = Haug's Target State (e.g. the alternation between ἵστημι and ἕστηκα).
Bybee's Anterior = Haug's Resultative = Traditional English Perfect
If the view of Bybee et al. (1994) of target state perfects (=resultative in Bybee's terminology) is correct, the propositional content of this perfect might convey the idea that Martha still believes Jesus is the Messiah despite her brother's death. Bybee argues that (her category of) resultatives collocate semantically with adverbs like still, while (her category) of anteriors do not. This might suggest that even with atetic predicates like πιστεύω, the perfect retains some of its historical/original function/meaning.
But that's just a guess.
Bybee, Joan, Revere Perkins, and William Pagliuca. 1994. The evolution of grammar: Tense, aspect, and modality in the languages of the world. Chicago: University of Chicago Press.
Haug, Dag. 2004. Aristotle's kinesis/energeia-test and the semantics of the Greek perfect. Linguistics 42, 387-418.
Canada Institute of Linguistics & Trinity Western University Graduate School