As the person who wrote that section, I'm sorry that it comes off as illogical. The thing is, the person who controls whirlpool decides when the dissipate token is removed in relation to all of the other upkeep effects that affect their objects. So, I can decide when to remove the dissipate token in relation to when I regenerate my unicorn, and I roll for the burns on my mage, etc. Or when Whirlpool damages MY creatures.
However, my opponent decides when it damages their creatures in relation to the other upkeep effects that affect their objects. So they decide when their creatures take damage in relation to when their troll regenerates, or their mage takes rot damage, etc.
Then, the person with Initiative decides when the dissipate token is removed in relation to their creature's taking damage from the whirlpool, because these effects are controlled by different people and you thus have a timing issue.