Is my water pump beginning to fail?

Even if we were to guess, you would need to describe the noise and where it’s coming from

 

Isn't that a youtube link in the OP?

 

That sounds to me exactly like the usual noise of the throttle actuator motor. Touch the throttle body with a mechanics' stethoscope (or just a long screwdriver with the handle pressed to your ear) and tell me if it isn't.

I would be wondering more why you hadn't heard it before in 5 years, but lots of things can make a sound more noticeable, or make us start noticing it more.

 

With that generation Prius, you can get at the water pump relays (in the rectangular box attached to the cowl at the back of the engine compartment). Watch some of the YouTube videos on flushing the coolant systems (there are two coolant systems, so there are two coolant pumps.) The videos I'm thinking about have you pull the relays and jumper the connection with an ammeter. This will allow you to run whichever pump you want while the engine isn't running. That way you can hear what the water pump sounds like, and you can see what current it takes to run the water pump. The videos will tell you what the normal current should be too.

 

I believe all these cars make the noise, it's to tell you the car is in READY. So you don't drain the oil thinking the car is off (I've heard mechanics do this)

 

It's not a noise made on purpose "to tell you" anything ... it's just that the throttle motor is PWM driven at a frequency people can hear, and the drive current is not shut off when the engine stops. (Maybe that decision not to shut it off then was the on purpose part to remind you the car's not off, or maybe it's just the way they happened to write the firmware.)

Honestly, whether you believe me or not, touching the throttle motor with a stethoscope or a long screwdriver to your ear will be about the fastest easiest answer to a question you ever got.

 

The engine is cooled by a belt driven water pump. If the engine isn't running, the pump isn't turning.

The inverter cooling water pump is located directly behind the driver headlight and runs whenever the car is in READY or if the car is in IG-ON (2 presses of the power button with foot NOT on brake). This pump is powered from the AM2 fuse, but can be disconnected at the single wire harness plug directly above and toward the passenger side of the under hood fusebox, or by unplugging it from the car harness right at the pump.

There is an electric water pump directly behind the inverter that provides flow to the cabin heating system.

There is also one more electric water pump in the US version that is mounted to the thermal tank (coolant heat storage) in the front driver side fender. IIRC, This should not be present in a non-North Amercian Gen 2.