I got similar message (TRANS. FAIL SAFE PROG) when I suddenly push fuel pedal too far for fast acceleration (just before kick down click). And then, the car goes accelerating very slowly.
However, when I stop engine & remove key, then start it again, msg disappears and does not appear unless I again push fuel pedal hardly. It never appears during grandma driving.
I checked my voltage (thanks to SlickGT1); found it 12.1 when engine switched off; ranging from 13.2 to 13.7 when running with lights/ radio/ AC conditioning.

What do you think guys?

E53 4.4, 2002, 265,000 Km