Think of it this way: In most phases and sub-phases, the game will ask each player "Do you want to do something?". It will always start with the active player (the player whose turn it is). If they answer no, the game asks the next player. However, if the active player does take an action, the process starts over. So, the game will start with the active player again. If both players answer no (pass), the game moves to the next phase/sub-phase.
I think the confusion is from Split Second. Normally, you could activate an ability "in response" to the active player's action and it would resolve before they were able to perform another (sorcery speed) action. But you were prevented from doing this by Split Second, had to wait until the spell resolved, by which time it is indeed the active player's action again.