OK. I get it now. I repeated the process, and then waited for the lens to dry. There were no spots. Instead of using distilled water, I used RO water from the system I bought at Home Depot and installed under my kitchen sink. It worked just fine.
This time, however, I let the lens dry after step 1, before proceeding to step 2. Using a flashlight, I could see that there was still dust on the lens. This is what I suspected. Rinsing with warm water alone did not remove all the dust. In my opinion, this is a flaw in this method.
If the lens still has dust on it, then rubbing the lens with a cloth in step 2 will scratch the very thin anti-reflection coating on the lens. Even though the cloth is wet, you are still pushing the lens cloth onto the dusty lens and rubbing the dust around the lens. The wet dust is abrasive, just like wet sandpaper is abrasive. The damage won't be noticable after one cleaning process, but it will build up over time with many cleanings. I recommend that you find a way to remove the dust before going after the greasy smudges, fingerprints, etc. This is essentilly what tulku described in his post when he used a wet Q-tip to wipe the lens with one pass.
If all I had was a lens cloth, I would very gently wipe the lens once from one side to the other using a wet lens cloth. Then fold the lens cloth in half and rub the lens to remove smudges. After that, I would discard or wash the lens cloth, because I won't remember later which side has the dust on it, and it's been thoroughly contaminated with oils from my fingers.