"....How are relaxed homeschoolers different from unschoolers? In my brief searches, they seem to be very similar. ..."They are and they aren't.
I've been both so I feel qualified to answer this.:)
When I was the eclectic/relaxed homeschooler my focus was still about making sure we were doing "educational" types of activities. Yes, we did mostly hands-on fun stuff the kids liked, but I was still seeing it as a way to touch on "science" or "history". We DID have a lot of fun but sometimes we'd hit walls where the kids didn't get all excited about something I planned and it made me all grumpy.
Not the best atmosphere for learning. I couldn't see that the video games they were fascinated with were more valuable than the homemade solar cooker I had planned.;)
I was always the most hands-on, relaxed parent of EVERY group we ever belonged to. I really wish we'd run into some real unschoolers earlier in the journey because I think it would have been a fairly easy transition from an academic standpoint.
The part that is missing with the relaxed/eclectic approach is still trust. It's just being more creative with how you get information into children,that's all. Better....but not quite the complete trust that unschoolers have in the human ability to learn.
We may still raise butterflies or garden or go to museums, so to an outsider we're doing some of the same activities. The difference is my kids can show zero interest and that is just fine. The difference is that I'm not trying to check of some subject box or define their learning experiences for them. The difference is that we do these things to have fun and trust that learning happens when we're alive and breathing.:)
My family started off as relaxed homeschoolers as well, then drifted into full-fledged unschooling, and our experiences sound so much like Ren's! Right down to our always being the most relaxed family of every group we attended. And it would have made a big difference in our journey towards unschooling had we met other unschoolers earlier on, as well... That's one of the reasons I'll happily answer questions over email and hang out with any new unschoolers who want to connect with families who've been doing it longer, as well as meet older/grown unschoolers who can be the "proof" they need that unschoolers do not have two heads. ;-) The other reason being that unschoolers are generally really great people to hang out with! :-)