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.
I don't think they are correct, close and understandable but not how a native English speaker would say it, I would say
- "He seems to not want us to help" and
- "He seems to want us to help"
negative questions are usually confusing so I'm not sure I can help you there.
- "It seems to not be working for me"
- "It doesn't seem to work for me" would be the same meaning.
- "It seems to not be working" would also be the same though applied to 'it' not just you working 'it'.
I think the difference is perhaps when spoken rather than written sometimes people miss/slur ot half say things as there is a lot more context.
Best Answer
When the adverb really comes before a negated auxiliary, the effect is of emphasising the truth of the sentence:
Here the speaker is emphasising that they honestly are unable to detect any difference.
However, when really comes after the negated auxiliary, the effect is usually quite different. Instead of making the meaning of the sentence absolute, it has the effect of making the commitment vaguer:
Here the speaker means that they cannot tell the difference in an absolute way. This gives the impression that they actually can tell the difference a little bit, but not in a way that is important or significant.
The Original Poster's examples:
In example (1), the speaker is emphasising that it does not matter at all. The sentence has an emphatic flavour.
In example (2), the speaker is saying that it doesn't matter in a major or significant way. They are implying that maybe it matters a bit, but not to a significant or meaningful degree. Far from being emphatic in character, the second sentence is slightly vague, and may be perceived as off-hand.
What causes these differences is whether the adverb really is changing the meaning of the word not— or whether the word not is changing the meaning of really (technically, whether not scopes over really, or really scopes over not). It's a bit easier to see with a follow-on sentence:
- It really doesn't matter. Really not.
- It doesn't really matter. Not really.