I'm back sooner than expected. I never ran any of the fixes from the HP link because another action(s) may have helped (which was recommended for another problem I was having).
So-
I had the printer connected to laptop via USB and Network connection. (cable wire between laptop and printer, and wirelessly through setup in printer) Due to lots of problems getting the "Auto Answer" light to stay on (i've posted on this separately in recent past), I have the Fax Settings arranged in such a strange way to make that work. So I didn't want to change those settings, which the advised link would have me do. For the "Auto Answer" problem, I ended up connecting computer to printer via cable wire. And that probably helped solve that problem.
I have kept both connections since then, and chose the USB-Wire (OfficejetJ6480 Copy1) as my Printer Preference setting in Printer Settings box under Control Panel. I believe that even when I made the Wifeless-Network connection active preference in Printer Settings box in Control Panel, the receiving fax or tests didn't seem to work (although this was some time ago and not tried just recently).
I think the duplicate connections was confusing things. The Fax Test seemed to pass when tested from settings button on the Printer. But it wouldn't pass when test was made from Fax Tools in Solutions Center on computer. (And this I have tested today.)
SO, I have deleted the Network-Wireless connection and "device" (officejetJ6480) from the Printer Settings box in Control Panel. I simply removed it. I have left the "OfficejetJ6480 Copy1" intact and keep the preference on for that "device".
I re-booted computer and ran Fax Test from both the Printer's Setup tool in printer, and from Solutions Center on computer. And they both show Passed Test. Go figure.
And also, the HPFaxMe test runs fine. So, 3 tests have passed. !
I hope this will be helpful for someone even as confusing it is.
But I'm sure one of the fixes in link from MissTeri must help too.
Thank you MissTeri!