Geeze, my brain is leaking on a holiday!
The printer is connected to my hard wired ethernet network.
I had to reset the printing system after the Mavericks upgrade. Then, re-added all printers.
The HP printer shows up by Bonjour, it can be found by IP as well (once I give the printer's IP).
Can't find it in fax.
I had to also install Gutenprint, as all drivers were gone after the Mavericks install.
It's looking like fax capability isn't present in Mavericks with that printer driver, which is rather annoying, as it was available in multiple other versions of OS X. I had to set up an emergency fax station on one of my Linux boxes, which is supposed to be dedicated for other purposes.
Searches found others reporting similar issues, one user ended up downgrading again. Apple support site pretty much says, see HP. HP says see Apple.
I'm no stranger to the command line at all, indeed, most of my system maintnance is done by command line, be it in a Windows system, Unix system or OS X system (Granted, it's a modified *BSD in Darwin, but it's different enough to give one a learning curve).
From here, it seems as if the driver either broke fax or doesn't include the fax funcitonality over the network any longer.