12V Battery Replacement Japanese Import 2000

Hello and Happy Holidays to all. This is my first new post and I'd appreciate any and all help.

I live in the U.K. and recently purchased a Japanese Import Prius from 2000. This means it predates the Generation I Prius built for the US/European market. All of the LCD readouts and buttons on the dash are written in Japanese...

My question relates to the battery. This vehicle has been driven for at least 2 years (probably many more) on a cheap mass produced standard 12v car battery (Halfords HB154 - £65). The battery looked very old and dirty as if it had been there for about 5 years. A month ago, when we purchased the vehicle, the power steering started having some serious vibration issues, even while parked, with the power steering completely failing on occasion. When the above battery died I replaced it with a new version of the same. Immediately, after a month of violent shuddering, the power steering worked smoothly and beautifully again. Knock on wood the car is running beautifully.

So, does the 12v battery's power output relate to the performance of the power steering?

Also, I would like to return the 12v I bought and replace it with the correct battery for a Prius.

So, what is the correct battery for an early Prius and why does the vehicle seem to perform perfectly well on a new cheap standard 12v car battery?

Thanks much