Avensis Washer System Failure

Sounds like a puzzle. Since both systems are down, a shared electrical component is plausible, like two light bulbs going out at once, suspecting the circuit breaker. It could be a fuse, relay, or even a bad ground connection. Have you checked the fuses specifically for the washer and headlight cleaning systems? Can you hear the pump trying to activate when you try to use either system?

I've checked the fuses, and they're intact. I cannot hear the pump trying to activate when I try to use either system. Is it possible to test the pump directly, and if so, how?

If the fuses are good and you're not hearing the pump, that narrows it down. Testing the pump directly is doable, but be careful! You'd essentially want to bypass the car's wiring and supply power directly to the pump. If you're not comfortable with electrical work, I'd strongly advise against it. Given the symptoms and your troubleshooting so far, it seems the cable connection of pump or relay is defective. A workshop will be able to diagnose it properly and replace the faulty component. It sounds like a simple job but could require specialist tools to properly diagnose. A repair like this would be about 165€

Thanks for the detailed explanation and advice. I appreciate you confirming the likely cause. I will take your advice and bring the car to a workshop for a professional diagnosis and repair. The estimated cost you mentioned is helpful. I will post the resolution.