There are two horizontal connectors in this illustration, one with four pins and one with six pins.
What are these connectors called?
They are called crimp housings. Harwin makes them:
and they are widely available. I get the 10-way ones and cut them down to the size I need.
They are generic "Rectangular Connectors" (....yeah). Digikey actually has a whole set of subgroups under the connector heading labeled "Rectangular":
Every manufacturer has their own line, with their own part number. However, searching for 0.100" (the spacing between pins) Rectangular connectors on a electronics component retailer's website should get you in the general area.