You had it right the first time: those are ZIF connectors. The white part that fell out is the latch you're meant to flip up in order to release the cable.
[https://youtu.be/WgcckmbziO0?t=2m18s|Here's a video of a tech working on a Spectre XT|new_window=true]. At around 2:18 you can see the ribbon cables getting correctly detached.
At this point, I'd leave the cable out and try to carefully reinsert the latch. Watch out that you don't damage the socket itself, or you'll be in for a much more complicated/expensive repair. With the latch back in place, you can slide the cable in and close it up.