Here's an even greater mystery for your files... It's the next day, now. I just looked down at my desktop. The Fast Dialer icon is back on my dock. Shifted one place, yes, but it's back there. I did not put it there, and the "360 Guards" app was not even set to active. (insert twilight zone soundtrack).
Oh. And the icon has a missed call notification bubble. That means something recognized it. (I set "360 Guards" in my bubble notification program, not even thinking this would work, since it's not a dialer app!).
In case I lose it again, and just to confirm I'm not imaging it, this is the elusive dialer program: