Unfortunately it didn't function successfully. I used this cable here at Amazon: https://www.amazon.co.uk/gp/product/B0013F5MAO
When I connected the Deskjet to the Pi via this cable, the nice system-config-printer GUI tool showed the printer and I could select the Deskjet 500 driver.
First I tried the default HPLIP driver, but the printer only printed some random dots on the left then. Then I tried the other driver, the Gutenberg one, and it didn't work either.
After some reboots and on-and-off-switching of the Pi and/or the Deskjet, I once got a correctly printed document, but it only worked once. When I tried to print a document again, it gave random dots again.
Maybe some Linux printer expert could solve this case, but not me.
Now I'm going to connect a Cannon (MP...) USB printer to the Pi and hopefully it will work. I know Cannon doesn't like Linux very much which is why I replaced my own Cannon printer with a HP one, but maybe it will work good enough for the casual computer user whom I'm trying to help. :-)
In any case, thank you all for your help!