As you can see, there are many ways to skin a cat (maybe not the best phrase to use for this subject matter).
I personally rinse the blade under a stream of water always angling the toe down so the water runs away from the pivot. Once the vast majority of the lather/hair is rinsed away I wipe the blade dry with a very soft short nap microfiber. I can usually shave half of my face or neck before seeing the need to rinse the lather.
After each pass, I set the razor on a small washcloth size folded up towel on the counter as far away from the sink as possible, then rinse my face (never tried not rinsing, but may experiment).
After the shave, I strop 15 laps on cotton, then 30 laps on leather followed by an alcohol wipedown. The last step is wiping the pivot and between the scales with a folded piece of toilet paper before storing open in my razor/brush stand on the counter until the next shave. Overkill? Maybe, but I enjoy the ritual and have never had a problem with rust despite living 10 feet from saltwater here in FL.:D