The 3-640429-4: The 0.156" (3.96mm) pitch matches the standard MTA-156 header footprint, and the single-row layout keeps the harness narrow for tight cable routing inside an enclosure.
It accepts 24 AWG wire, either discrete strands or ribbon cable, and the closed-end feature means the wires bottom out at a fixed depth so you don't over-insert. You'll need a hand or bench press tool to seat the cable; the MTA-156 IDC tooling is common in harness shops.
The receptacle mates with any standard MTA-156 header — through-hole or right-angle, 4-position, 0.156" pitch. The white housing colour is standard for MTA-156 receptacles, making visual inspection of the latch engagement straightforward under a work light.