Specifically, the book says it starts at 0 and increments each time an event from the list happens, one such event being "the adventurers enter a new level of the dungeon." Does the very first level of the dungeon count for this? As in, is the submergence die set to 0 before the party goes into the dungeon, and therefore the act of entering the first level advances it to 1; or is the submergence die set to 0 once the party is in the dungeon, and so it'd take going to level 2 for it to advance?
Yes, according to RAW.
On p20 we find that:
The submergence die starts at 0. Increase the submergence die by 1 when: The adventurers enter a new level of the dungeon.
The characters are running around above ground, the submergence die is 0. The find the Maw and decide to go in? The submergence die is set to 1.
Because of any of the following reasons:
- If you don't see the Maw as a level of the dungeon
- you want a slower start to dungeon exploring
- you just started your once-a-month-dungeon-session
- you don't like the RAW
then the submergence die is 0 at that moment. If they then go deeper to the Gauntlet or the Gizzard (or some other level), you increase the die by 1.