While trying to detach the cable, I pushed out the black part, which came out from the opposite side of where the cable enters the connector.
The cable was previously tightly clipped in place and came loose after the black part came out.
How do I re-attach the cable, or is the connector permanently broken?
Is there a name for this particular connector? There seems to be a few names, like zif, etc but I'm pretty sure mine isn't zif.
That black part is a wedge meant to secure the flat cable strip. Attach the strip, then reinsert the wedge carefully and you are ok.
Just go to distributor site and look for a replacement part if you want to replace it, your connector might be a backflip type or clamshell. Count the pin numbers and measure the pitch to make sure you get the right one.
Or if you are looking for a quick fix, you can try to insert the cable into the slot, make sure it's firm, and then use a kapton tape to hold the cable to the PCB surface. I've done this when we can't wait for a replacement part.
ZIFmeanszero insertion force.... a zif connector is any connector that has a clamping mechanism that allows the cable to be "dropped in" without any pushing \$\endgroup\$