I think 3-way backlinks happen when Site A links to Site B, and Site B links to Site C, then Site C would link to Site A.
Exactly right. The reason why webmaster utilize this method of linking is because it is believed that the search engine spiders will treat each link as a one way, inbound link to each site, thus carrying more weight that reciprocal linking.
This is called triangle linking and I read somewhere that Google is onto this trick and now may be penalising for it. I suppose the next progression from here would be "link rings" involving more than three sites. Kind of like those webrings back in the day.
3-way backlinks make when Site A links to Site B, and Site B links to Site C, then Site C would link to Site A. it is the way to make the back link