For new file save speed run, is it required to only beat Praxis once? Or should it be required to beat the game on Doom difficulty?

If the later, it'll be required to do no less than 2 runs. Personally, I'm in favor of requiring 2 runs to trigger the credits.

Now that the game has been released, Doom should be split into 2 different categories. Doom only requires that 15 modifiers be enabled (This is also the condition that triggers the end game credits and should be considered full game completion). If all modifiers are set to Max, that is called a DEMONS run.

DEMONS conditions are:

All difficulty modifiers on max. No Nexus shop purchases. On version 1.0 (or later.) Recording on video.