I think the confusion stems from small, independent presses calling themselves indie presses, and independent individuals calling themselves indies.
Lots of self publishers that I cross paths with here and there call themselves indie, because they do everything themselves. They are doing the thing independently of anyone else. They call themselves indie published.
It's sort of like the confusion between trade and traditional, and people using the weird term 'trad publishing.' Which hits the ears funny, but most people grok it.
Then there's the label of vanity publishing, which might refer to a vanity press (which you pay, and so they make their money off their authors not their sales) or might refer to self publishing, which some writers disparagingly call "vanity publishing."
Lately, some self publishers say true vanity lies with those who insist on a trade deal, who refuse to get their work out in any other way... it's vanity, the argument goes ... thereby bringing this particular term full circle, heh.
But "self publishing" is clear.
I self publish.
ETA. Oh, and then there's hybrid. Which means different things, too.