So maybe it's a problem with the printer's IP?
Yes. As far as your network is concerned, the printer is just not there.
You will not be able to progress until you assign it an IP address accessible to your LAN. Apparently it needs to be 192.168.1.x (a 1 in the place that is now a 5), where x = a large enough number that is not likely for router to assign to other equipment.
Your challenge is to assign the printer that IP address. Let's use the address 192.168.1.150.
HP's support pages for that printer and its Jetdirect card say, rather unhelpfully:
Using the front panel of printer
The IP address may also be changed using the front panel of the HP Jetdirect device. This method of front panel configuration varies greatly depending on the model of HP printer that is being used (consult the User's Guide for the front panel instructions for a particular printer). Click here to go to the HP Web site. Use the Search and search for Front Control Panel TCP/IP Configuration .
I did that and it didn't really get me anywhere, since there seem to be few if any references to the non- "MP" or "V" etc. variants of the original LaserJet 4. So, try to find a setting in the printer's front panel that bears any resemblance to "network settings" or "IP address".
If you are able to navigate its front panel to a place where you can enter the IP address 192.168.1.150 be sure to press GO or ENTER or ONLINE or whatever you think might make the setting "stick" after you enter those digits. Print the same test page you did before, and ensure its IP address has really been changed. Then, then try the ping command again, with its new IP address:
ping 192.168.1.150
If you are unable to determine a way to use the printer's front panel to change its IP address, you should be able to use Terminal to set it. This will get a bit cumbersome so use it only if you are unable to change the printer's IP address some other way:
Terminal is a program in your Mac's Utilities folder.
Locate Terminal and open it. At the command prompt, type the following exactly as written (copy and paste the following to avoid errors):
sudo arp -s 192.168.1.150 00:60:b0:91:21:e7
... and press Return. You will get a warning of impending disaster should you make a mistake, and requesting your Mac's Administrator password to proceed. Type your password (which will not appear as you type it) and press Return. Unless something went wrong the command prompt should immediately reappear as if nothing happened.
Next try the ping command again:
ping 192.168.1.150
You should receive ping responses from the printer as in my earlier example.
If you do, then congratulations because you have made it all the way to Step 1 😮
You still have to add the printer in System Preferences > Print & Fax, using this new IP address, and make sure the Gutenprint driver works.
Edit: The reason your printer works with your old G5 running Tiger is that it is using AppleTalk for communication. AppleTalk doesn't exist in Snow Leopard. This is the reason you must use IP for communication on your Mac Pro. Your old G5 should continue to work with the printer as it always has.
another Edit: HP's support page alleges that the Terminal command arp above will only work with JetDirect firmware versions x.08.03 and later only. Yours appears to be A.05.05, so the front panel option may be all there is. If that does not work I have one more idea but it's an even bigger pain.