Help Identifying Small 4 pin male connector
That connector looks very like a JST XH connector

The pitch is 2.5mm.

The JST web site can be searched for 'XH' here 
Or the JST XH series is here.

There is a link to a downloadable PDF but it isn't clear that it is stable for the long term, so this link may break.

The engineering drawing "Through-hole type shrouded header", page 3, is 7mm high. 

The datasheet has another drawing which says 9.8mm high on page 1, but that shows a plug in a socket, that drawing also shows 7mm as the height of the socket