It looks to me like Stage 1 works as follows.
The 128 players are split into 4 groups of 32, playing single-elimination, alternate-break races to 7. The 4 winners from each of those 4 groups win a place in Stage 2.
Then the same 128 players (minus the 4 who have already qualified) are again arranged in 4 groups of 32, and the single-elimination races proceed again. This is the "2nd chance" part of Stage 1. Again, 4 winners emerge, to be placed into Stage 2.
Then the 128 players (now minus the 8 who have already qualified) are again arranged in 4 groups of 32, and the single-elimination races proceed for a third time ("3rd chance" part of Stage 1), resulting in the final 4 qualifiers for Stage 2.
So Stage 1 is essentially 12 32-player single-elimination tournaments, with each of the 128 Stage 1 players slotted into 3 of the tournaments.
Edit: An alternate way to view Stage 1 is that it is 3 consecutive 128-player (or, really, 128, then 124, then 120) single-elimination events (1st, 2nd, and 3rd chance), with the last 4 players in each event qualifying.