I don't have that design posted anywhere. Here's the circuit diagram in image format:

I don't use Arduino boards but rather the microcontroller chips directly, which turns out to be pretty simple to do (here's some videos on doing Arduino stuff that way).

Because I'm using discrete components (mostly DIP so that they can go into a perfboard but also some cheap SMD MOSFETs because low power SMD MOSFETs go for 2 cents each and work fine for just driving a small water pump whilst all the DIP MOSFETs are power MOSFETS, so much more expensive and way over spec for the kind of currents a small water pump pulls) so it all adds up to very little cost, plus I made a board design for it and had 20x of that done by some cheap chinese PCB maker so even the board ends up even cheaper than a perfboard.

Doing the whole thing around an Arduino board, even a cheap Arduino Micro is much more expensive, way over-spec for an automated watering device plus those boards tend to eat up lots of power when in sleep mode so the batteries wouldn't last long if the circuit was built around such a board - if you look at the diagram, all the paths from VCC to GND either go via the microcontroler (like how SIGNAL- and SIGNAL+ drive both sides of a two color bi-directional LED) or have a control MOSFET that blocks by default when the micro-controller doesn't send any signal (like SENS_CTRL).

The $3 bucks are a rough estimate but I wouldn't be surprised if it's an overestimate. It really depends on the price of the capacitive sensor you're using (I just get mine from Aliexpress) as that might actually be more expensive than the rest. However do note that the price of the water pump was not included and that adds maybe another 2 bucks.

EDIT: For some weird reason I kept mentioning "diode" when talking about MOSFET transistors, so I corrected that.