Replacing the oil screen cover gasket
To start, pull off the oil strainer cover (#14090-1350). Make careful note of the five bolt locations, as they are of varying lengths, as explained in the above-mentioned article. Clean the screen as directed, then check the gasket. If it looks good, with no rips or noticeable wear, put everything back together and ride on. This is not something that will need to be replaced very often.
This paper gasket (#11009-1874) was worn on the edges, so the decision was to replace it. This type of gasket has a tendency to stubbornly come off the cover in very small pieces (it's paper). Use a flat head screwdriver or putty knife to remove the rest of it. Be patient and work slowly; any dings or gouges in the cover could result in an oil leak.
45 minutes of careful scraping and cleaning later, the cover surface is finally clean and clear of the old gasket.
The next step is to take the new paper gasket (part #11009-1874) and orient it correctly.
Before you button everything back up, there is something you can do to increase the life of your paper gasket. Coat the outside of the gasket (the cover side) with oil and leave the engine side dry, so it'll stick to the engine. This will make the cover come off more easily. This can be done with any paper gasket. Coat one side with oil; the other side will stick to whatever it's facing.
Go ahead and reassemble the screen and cover, finish the rest of your oil change, and be sure, as always, to check for leaks.