Yes, that's right, tzfujimino. I like how you've put it in terms of "gaps"; that keeps things fairly simple.
Things get more "complicated" -- and more controversial -- when we talk about things like wh-traces inhabiting those gaps, such that the reason contraction is blocked in (2b) is that a wh-trace is sitting there blocking it, like syntactic dark matter.
It was George Lakoff who, in 1970, noticed this phenomenon first. (But the original discoverer may have been Lawrence Horn.) That was before trace theory. His explanation was that "
want to cannot contract to
wanna if there is an intervening NP [noun phrase] at an earlier point in the derivation." His famous examples were these:
(3a) Teddy is the man I want to succeed.
(3b) Teddy is the man I wanna succeed.
Sentence (3a), he observed, is ambiguous. It admits of both an altruistic reading, on which it relates to "I want Teddy to succeed," and an ambitious reading, on which it relates to "I want to succeed Teddy."
But (3b) is unambiguous (for almost all English speakers); it only has the ambitious reading, because the impossibility of
to-contraction across the gap or trace rules out the possibility of the altruistic reading. The explanation has been controversial, though. Pullum attempted an alternate explanation in 1997 (see
here).