What’s the correct way to disconnect this circuit board connector?
As MadHatter also suggested, I also think this looks a JST connector, maybe from the SH connector series.  

I unmount these SH connector using a sharp tweezer or two screwdrivers and gently pry loose the house from the header such that the house leaves the header as straight as possible.
I position the tweeser points between house and header on both sides as shown in the picture below (between the red lines on each side).  



While gently prying the house one side, I try to fixate the tweezer pijt other side on the header, such that it acts fulcrum. Alternating this per tweezer point, I gently lift up the house bit by bit till it comes out.

For clarity, I drew and explained the process  using the long side of the connector.  Prying at the small sides of the connector might even go easier.

I think this method is always better than pulling on the leads of the connector.