What is the name of this connector? (4-pin with a vertical line)
I think it might be MOLEX 5023860470 - 1.25mm Pitch CLIK-Mate connector (or some compatible one) .

Here's datasheet: http://www.molex.com/molex/products/datasheet.jsp?part=active/5023860470_PCB_RECEPTACLES.xml

And link do Digikey: https://www.digikey.com/product-detail/en/molex-llc/5023860470/WM4948CT-ND/2700570