Name of slightly-less-than-2mm pin connector?
Those are called RMC female connectors.

I think this link shows the connector you are looking for, but you should find a 7-pin for yourself.

http://www.rhydolabz.com/hardwares-connectorssockets-c-170_77/4-pin-rmc-female-connector-with-wire-2mm-pitch-p-2345.html