I think they're correct BUT merely indicative/for gearwankers, just like the same total watts figure on an ordinary all-in-one power supply...

At least the classic picopsu (small card with motherboard connector directly attached on the side, not sure about the bigger ones that are about as big as a first Xbox power supply) merely passes through the 12 V so that's a prime way to inflate the ratings (160 W ≈ 5V8A + 3.3V8A + 12V8A), at this time 12V was mostly for the motors so by going solid state you have almost no load on it and the "160 W" are already down to 66 before accounting for the fact the load will not probably be even split between the 5 and 3.3 V lines…

…and then you have to deal with the claim, in the manual, that those twin 8 A are only with active cooling!

For whatever it's worth, my HP Vectra VL400 DT came with a 120 W power supply (more traditional and certainly more balanced for this system, with an 800/133 processor, 2 sticks of memory, 5400 RPM disk, FDD, DVD, and a Radeon 7000 it was stable but at limit (didn't start when cold, always needed a second push of the button)
When it broke, I replaced it with (custom metalwork and) a Power Man (InWin) IP-P300EFT-2, which despite being a modern FTX brick for modern modest PCs is still generous on 5 V output (OK, I bought it because it cost 10 €, I noticed this later 🤣)

As for your final question - unfortunately, pentium 4 is probably the ideal case for a picopsu (CPU regulators and motors on 12V, most logic on 5, most cards on 3.3), not a high end system from the previous generation 😀 🙁

(The 4 pin CPU power cable is commonly called a P4 connector, not sure if this is relevant to the processor with that name since I've found power supply connector names to be all over the place if there at all, but apart from the mostly failed "aux" connector a few years before your computer, that was the first upgrade on base ATX)