to put to death: to end a life on purpose, whether by a court's sentence, a betrayal, or a killing blow, and by extension to kill off whatever still holds power over a person
This verb always has a killer in view. It is not the word for simply dying (ἀποθνῄσκω), which describes the person who comes to the end of life with nobody named as the cause. Here someone decides on a death and someone else suffers it. In the Gospels that decision is usually formal: the chief priests and the council hunt for testimony that will let them put Jesus to death, and by morning they have agreed on it.
The same verb also covers deaths that are not official at all. Jesus warns that families will hand each other over and that children will have their parents put to death, and that some of his followers will be killed by their own relatives. Paul uses it for the daily danger the apostles live in, quoting a psalm about being killed all day long for God's sake, and for the fact that though they are punished they are not actually killed off. Peter uses it of Christ himself, put to death in the body and made alive in the spirit.
Paul then turns the word inward, and that is where it separates from the plainer verb for killing (ἀποκτείνω). Believers are told to put the body's practices to death, and are told that they themselves were put to death as far as the law is concerned, through the body of Christ. In both places the picture is an execution rather than a slow fading: something is deliberately ended so that something else can live, and the harshness is the point.