Colorbox grouping content linked from same page - how to stop it?
I figured this out, with crucial help from the #Drupal IRC channel. 

I was using Colorbox Node along with Colorbox, and I was confusing the documentation from the two of them. Specifically, Colorbox requires links to have the class colorbox-load while Colorbox Node requires that they have the class colorbox-node. The really confusing thing about this is that if you use colorbox-load instead of colorbox-node, it works, sort of. It was loading the entire page including headers, footers, and all markup and styles, into the modal, as if it were an iframe. 

So step one was to start using the correct link class.

After changing the link class to colorbox-node, however, all I saw in the modal was the title of the target page, unstyled, on a white background. Poking around in the Colorbox settings was, unsurprisingly, unhelpful. Again, #Drupal pointed me in the right direction: Colorbox Node creates a display of the content type and makes the body hidden by default for some reason.

So step two was going to structure > content > [content type of target node] > manage display and choosing "Colorbox" (next to "Default" and "Teaser") to unhide the body of the target node.

Voilá, you now have the title and body pulled out from your target node and appearing isolated in your Colorbox modal.