It really varies - a bit like people's definitions of part and full livery!
I've been on various guises of Assisted DIY from services available on an ad-hoc basis to pretty much a basic 5 day part livery package but owner supplies hay, bedding and feed.
At one time I was on Assisted DIY which involved Mon-Fri feed, turn out, skip out and put up evening hay, bring in and feed (with rug changes available for extra £) and my friend was on a part livery package on another yard where they had to muck out so it shows how varied the term can be!
If you're offering it or looking for it - be clear in what you want to offer or need to avoid any misunderstanding is my advice