Comparing negatives: "she seems not to know" vs. "she doesn't seem to know"

Solution 1:

Yes, there is a name for this kind of alternation between constructions.
It's called Negative-Raising, or Neg-Raising (NR), among other things,
and it's governed by the predicate seem in this case;
there are a number of other predicates that govern it.

NR is a minor cyclic alternation rule.
That means that it is governed by the matrix predicate (all cyclic rules are governed)
and that the set of predicates governing it is small and specialized (that's the "minor" part)
and that it relates two different but synonymous sentence structures (that's the "alternation" part).

What happens is that, when you have a complement clause with a negative in it, like

• Bill wanted/seemed/intended/tried/managed not to be driving the truck.

with some predicates, but not others, this construction is equivalent to the same
sentence with the matrix predicate negated, but the complement not negated:

• Bill didn't want/seem/intend to be driving the truck. (equivalent with want, seem, intend)
• Bill didn't try/manage to drive the truck. (not equivalent with try or manage)

I.e, want, seem, and intend govern NR, and try and manage don't govern it.

Another way to look at it is that NR predicates are "transparent to negation", because
they don't really contribute much to meaning beyond individual perceptions and desires.
Whereas the vast majority of complement-taking predicates do contribute to meaning,
and are therefore "opaque to negation".

By the way, I used infinitives in the examples above for simplicity, but the phenomenon
is not limited to them. With the right predicates, tensed complements can undergo NR, too:

• I thought (that) you didn't want toast. = I didn't think (that) you wanted toast.
  but ...
• He said (that) you didn't want toast. ≠ He didn't say (that) you wanted toast.