I would guess that when there's a problem, there are a lot of people working concurrently, all of who have access to the data.

I would guess that someone who is responsible for monitoring systems on the ground and isn't at risk of losing oxygen and blacking out and isn't having to get into a spacesuit is probably the principal person involved with assessing the condition of the station in an emergency.

I'd also expect that the person who makes the big calls is on the ground.

searches

It sounds like there is a person aboard the ISS who has ISS command, but normally, it's a person on the ground who makes most of the calls:

https://en.wikipedia.org/wiki/List_of_commanders_of_the_International_Space_Station

This is a chronological list of commanders of the International Space Station. A pre-decided inhabitant of the ISS assumes command upon departure of the previous commander, at the end of an expedition, in a small hand-over ceremony. Their responsibility is defined by the ISS Code of Conduct, which states that the ISS commander has some authority over the operations of the ISS, but should ultimately defer most decisions to the Flight Director.

https://en.wikipedia.org/wiki/Flight_controller#Flight_director

Flight director

Leads the flight control team. Flight has overall operational responsibility for missions and payload operations and for all decisions regarding safe, expedient flight. This person monitors the other flight controllers, remaining in constant verbal communication with them via intercom channels called "loops".

Apparently the (older) Russian modules had been leaking air. The leakage rate was manageable, but the cause of concern was that the total rate of leakage abruptly doubled. Might be that they were worried that there might be some kind of progressive failure occurring.