Which of these deductions make sense?
The simulator is actually being run by the adversary. It makes sense that the adversary initializes the simulator (which is just a program) with the input that the corrupted party (also just a program) will use.
The adversary is actually being run by the simulator, and the simulator (which is just a program) will give the adversary (which is just a program) the information it needs.
We just should consider the adversary as a black-box that the simulator has an access to it and vice versa.