First, "seldomly" barely exists: the OED marks it as "obsolete" and the iWeb corpus shows 611 instances, against 759 589 of "seldom". Seldom is the usual adverb.
In answer to your main question, attempting to rank these precisely is a waste of effort. Language is used by people, not machines, and they don't necessarily have hard and precise rules. Sometimes, occasionally, rarely (which you don't list), seldom, and hardly ever can all overlap.
I would say that the main difference between occasionally and seldom/rarely/hardly ever is that occasionally has positive polarity, but the others have negative. This is not obvious from the definitions, but it has implications not just pragmatically but also on syntax.
If you ask somebody how often something happens, and they answer "occasionally", this is usually neutral, not implying any expectation.
If they answer seldom, rarely, or hardly ever there is often an implied "contrary to what you might have expected" (or "contrary to what I might have expected").
The syntactic effects of negative polarity are well known. Two of the most salient are:
- They take any and its relatives rather than some:
I have occasionally seen somebody there. (not anybody)
I have seldom seen anybody there. (somebody is possible, but less likely)
- Negative polarity adverbial phrases can come clause-initially, triggering subject-verb inversion:
Never/rarely/seldom/ have I seen such a thing.
But not *Occasionally have I seen such a thing.