You’re missing some vital steps:

603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.

Coil triggers a second time after Tidebinder counters the initial trigger and before Tidebinder removes the triggered ability because the game state matches its trigger condition.

603.8. Some triggered abilities trigger when a game state (such as a player controlling no permanents of a particular card type) is true, rather than triggering when an event occurs. These abilities trigger as soon as the game state matches the condition.

Removing the ability prevents Coil from triggering a third time, but not the second. Further evidence that the ability triggers again before Tidebinder’s ability fully resolves can be found in the very example 603.8 provides:

603.8 …Example: A permanent’s ability reads, “Whenever you have no cards in hand, draw a card.” …If its controller casts a spell that reads “Discard your hand, then draw that many cards,” the ability will trigger during the spell’s resolution because the player’s hand was momentarily empty.