Are these pin connectors a standard part? If so, what are they called?

Those are called Bootlace Ferrules and, yes, they're pretty standard in electrical wiring. They come in many different sizes, and each size has its own colour.

There are also Twin Entry options, that allow two wires to be joined:

The pins are bootlace ferrules.

The pins do not stick out from the plug (see 3rd image that shows the main pins on the PCB).

If I'm reading your photos correctly the wire ferrules are held in cage-clamp terminals on the PCB connectors. The ferrule pins do not stick out the other side of the connector.

Figure 1. Wago cage-clamp terminals.

These terminals can grip the ferrules or stripped stranded wire. They have become very popular as they are quick to install and are vibration-proof. The wet-sensor manufacturer probably uses them to prevent problems with stray strands, etc.

Figure 2. Cage clamp and wire showing excellent contact area.

Cage-clamp terminals actually work better on stranded wire than they do on ferrules. In Figure 2 you can see that the strands have formed a rectangular bunch and that contact is made between the terminal and many individual strands. When a ferrule or solid wire is inserted the contact will be tangential at top and bottom.