In the 4 tetrads of Anipanasati (a map of breath meditation), item #3 and #7 are to be sensitive to the body and mind. This can be seen in multiple ways.
Sensitive first is just listening, observing. You don't disturb what is going on. You try to do very little and see what has momentum, what keeps on moving because it still has energy.
Sensitive second could be watching the momentum change. In the widest sense, you can watch to see where a stirring (of sadness, happiness, wanting, or aversion) lessens and lessens until it goes away. If you are lucky, you will see several circuits of this so you start to see the pattern. If you are even luckier, you will see how your habits (old karma) might shoot off a thought or feeling that keeps a stirring to stay stirred up (it gives it more energy).
Sensitive third is a level of gentle experimentation. You can try to adjust (slowly, carefully, with lots of noticing) your perceptions or thought patterns. You arent trying to overpower it... In general everyone can always squash a stirring if we apply lots of effort; it isn't a rare or great skill. You are trying to do the smallest things possible to see what then changes in the mind. Maybe taking 3 deep breaths. Maybe just saying, "what am i choosing". Maybe a question, "am I keeping this stirring/sensation going?" And then watch. It is like a careful chemistry experiment where you did a small thing (input) and are trying to notice the result (output).
All of this is sensitivity of the body (#3) or of the mental qualities (#7).
Only then, after you have been sensitive, that is where the Buddha lists "calming". This is after you have looked at all your tools of thinking. And then you decide what tools of thinking to apply.
Calming is different from ending a stirring. In an important way, we want to make sure the tools we apply to end the stirring doesn't create another stirring. A major new stirring happens when we bottle it up (anger, desire, spinning thoughts). When bottled, the pressure can build. The cause of anger or greed or spinning thoughts, if it remains, keeps building it up. If this is true, this is where we go back to sensitivity. One might say, "i have a level of calm right now, but i see my actions creating a bomb in the future."
We have to notice and to be honest with oneself. We might not know if we have calmed ourselves with a future bomb or without building pressure and thus avoiding a future bomb. Is this case, we might need to be very cautious and resolute, checking (yes, checking is allowed) to see if pressure is bulding up. If it builds up in a way we are familiar with, we can address it... relieve the pressure or cut off the source. But it could build up in a hidden way or unfamiliar way. In that case, we should be quick to react (like a firefighter), but we cannot beat ourselves up for not seeing. We can push ourselves to do better next time to check. With gladness.
The process of calming is a tricky thing. It is not enturely straightforward. It is a bit like whack-a-mole. Knock mole #1 down and mole #2 might pop up. And in tricky ways in our blind spots.
So we need to be very careful. Don't skip the step of noticing and being sensitive. When we are very sensitive, we can generate calmness in a full and benficial way. When we are not sensitive, generating calmness can have impatience and a lack of skill. And it CAN cause more harm then good. In the short run, we can create other stirrings. In the long run, we can get confused and think that calming IS the goal, we can aim too low and focus on the surface elements of the mind.