Lube the cylinder...the piston doesn't need it...
Apparently no one has bothered to tell the people who write the Jabsco manuals that petroleum degrades the rubber parts in toilets...or maybe they DO know and their advice to use it is part of a plot to make sure their toilets need to be replaced early and often.

But more likely they were written by the same guy who came up with this gem: "use only ethylene glycol antifreeze because automotive antifreeze is destructive to rubber." (If you don't get it, I'll explain it).
To lube the toilet (I do indeed recommend thick teflon grease):
While you have the top off the pump, Stick the tube nozzle into it and give it a healthy squirt....pump a few times to spread it all over the inside of the pump...put the top back on...you're done.
If you'll do this every year as part of spring recommissioning, your toilet will pump smoothly and quietly all season without the need to do anything more, and your seals, o-rings etc will last about twice as long):