Electronic – Help identifying a M12-like 4-pin connector with 6mm diameter

Those are headers and receptacles.

The parts in the pictures look like the common 2mm battery connectors made by JST. If those connectors will work then I would search for JST and 2 pins and 2mm pitch. The exact pn's for the TH parts should be

• B2B-PH-K-S for a top entry header
• S2B-PH-K-S for a side entry header
• PHR-2 for a cable mount recptacle (without contacts) The contacts are SPH-002T-P0.5L for the 24-28AWG crimp

You may want to double check those pn's at Digikey and Mouser.

Connectors are one place where all the search engines fall short. The printed catalog can sometimes be a lot quicker.

As @Nick Alexeev said in the comment, it's propably a Molex PicoBlade.

http://uk.farnell.com/molex/53261-0471/wire-board-connector-header-4pos/dp/2411457

Can't be sure only looking at your picture. If it would be me, I would get a plug that fits into your controller since you can see exactly what kind of connector you need from the controllers datsheet and replace the small connector from the fan. Soldering wires is not a bad thing either but wires have a habit of breaking off pretty easily. Hot glueing around the solder is a cheap man's trick thats not pretty but works.