Yes, the expressions “moving” – “not-moving” are rather misleading.
The actual distinction according to this theory (or to my favourite flavour of this theory) was: “able to act willfully” – “unable to act willfully”. The distinction was quite logical. Cp.:
(A) “Susan runs down the hill.”
(B) “Water runs down the hill.”
(C) “Susan drinks water.”
A speaker of Proto-Indo-European might have said, learning English: “This is strange – as if these were somehow the same acts or events. ‘Susan runs’, OK, but water does not run – it is dragged down. It does not act, something happens to it.” (And then the PIE-speaker might have speculated that anglophone people have an animistic worldview or so.)
In PIE the subject of a sentence seems to have denoted an actual agens; and there seems to have been a class of nouns for things unable to act willfully, hence unable to be the subject. So that the subject in (A) and (C) would have been a subject in the nominative, denoting an agens, whereas in (B) “water” would not have been seen as subject; the sentence would have been constructed differently and “water” would have been put in the same case as “water” in (C).
Because all over the old IE languages, neuter nouns have the same form for the nominative and the accusative, and it seems to have been rather the form for the accusative: Cp.
Latin masc. nom. sg. ‑us
Latin masc. acc. sg. ‑um
Latin neutr. nom. sg. ‑um
Latin neutr. acc. sg. ‑um
There is further evidence, too.
In principle, the distinction was quite logical, but its application is difficult and gets easily messed up over time until the old semantic distinction has become a mere formal one.