So yeah, the long and the short of it is that in season 2, the clock does not re-start at 0, it resumes wherever the player left off.
You'll notice that the differential for one season is 10 days, but teams always wait 30 days in the player's rookie season, it's because of the way the arb clock works the teams are lumping the entire 30 day service time "waiting" gap into the first season instead of spreading it out with 10 days over each of the player's first three seasons, because that makes the most sense (30x1 in the player's youngest season > 10x3 which would include time in following seasons when the player has gotten better)