What is the pin pitch, and what type of plug or cable fitted in to it?
They look to me rather like solder pin connectors that were made to be permanently crimped to FPC or ribbon cable, to allow them to be soldered to the PCB in mass production.
Not the same exact part as these, but a similar concept?
Connectors tend to be really difficult, so I can't identify it, other than how it works. It's a ribbon cable socket, and the top part pulls up (as in the lower socket), so you can insert the ribbon cable - you then push the top part down (as in the upper picture) to lock the ribbon in place. To remove it, you simply pull the top part up again.
As rjenkinsgb said, they look very similar to the permanently crimped ones - but it's not one of those (thanks to the good picture) - I used to come across them all the time on LCD TV's, particularly when replacing LCD panels. Sometimes instead of the push-in types like yours, they did use permanent connectors at one end - and just to keep you more confused, there was an alternative type where you pushed the top down to release the cable, and the top was spring loaded.
They look to me rather like solder pin connectors that were made to be permanently crimped to FPC or ribbon cable, to allow them to be soldered to the PCB in mass production.
Not the same exact part as these, but a similar concept?
That's close, but, as Nigel Goodwin correctly described, the connection is not permanent. I've been unable to locate this kind of connectors at digikey, mouser, etc. Looking for specific technical terms to search by if there are any.
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It seems this kind of connections has been superceded by FPC in the industry. I think I will go with a more ubiquitous type, such as JST PH. Thank you, rjenkinsgb and Nigel Goodwin, for the helpful responses!