There's some confusion over these terms. Indie publishing sounds cool, whereas self-pub has a stigma going back years. It's no surprise many self-pubbed authors are calling themselves indie published.
I resisted at first because self-pub is what I've always called it. You're publishing your own work. Nobody else is doing it for you.
Then I saw a post which brough up a good point: is it self-pub when you pay for cover art, editing, proofing and layout? Those people are doing a professional job, and it seems a bit odd for the author to claim their work under the 'self-pub' banner.
There's another point of view as well. If you set up your own imprint, purchase a block of ISBNs and go into business, this entity is still technically a publisher even if it only exists to publish your own work. Bowker and Lightning Source certainly think I am.
That's the stage I'm at now: I have my own imprint and I hired pros to get my book ready. I should probably stop calling it 'self-published' and start calling it 'indie-published' instead.
Simon Haynes is the author of the Hal Spacejock and Hal Junior series (Amazon / Smashwords / other formats)