Usage of hain't

This may not be confirmation, but Mark Twain, who has this to say about his dialectical decisions in The Adventures of Huckleberry Finn, uses both words following the distinctions you have made.

He uses ain't solely to me "am not", "are not", or "is not", as in:

"'Stuff! stealing cattle and such things ain't robbery; it's burglary,' says Tom Sawyer. 'We ain't burglars. That ain't no sort of style. We are highwaymen.'"

By comparison, he uses hain't to mean "has not" or "have not", as in:

"'I hain't ever done you no harm. You know that. So, then, what you want to come back and ha'nt me for?' I says: 'I hain't come back—I hain't been gone.'"

Wiktionary claims that in certain dialects, hain't and ain't are synonymous.

Hain’t originally derived from han’t, and meant "has not" and "have not". In certain h-adding modern dialects, hain’t is synonymous with, and a replacement for, ain’t in all its uses.

However, if you're writing or speaking in a dialect that uses these variations and you plan on employing both, interchanging them may cause confusion.

Hain’t is attested by over 30 citations in the Oxford English Dictionary. A typical example from 1878 is ‘Nobody can say she hain't been a good yoke-fellow; she's kept up her end’, while another from 1898 is ‘I hain't ben runnin' the Eagle tavern fer quite a consid'able while.’

I would imagine that this pronunciation of ‘ain’t’ arose from uncertainty over whether or not to pronounce /h/ at the beginning of words that begin with a vowel. Those two citations suggest a London dialect, but I have never heard it myself anywhere in the UK.