therapeuō

This is the meaning the word carries almost everywhere in the New Testament. It describes making a sick, disabled, or demon-afflicted person well again. It is the standard word for what Jesus did as he went through the towns of Galilee, and the word he handed to his disciples when he sent them out to do the same.

The healing in view is real and physical: blindness, paralysis, fever, chronic bleeding, every kind of disease and disability. The same word covers being freed from tormenting spirits, because in the Gospels that release is told as a kind of cure as well. When the word appears in the passive (someone was healed), the point is that the person actually got well, not merely that they were treated.

The sense is steady across all four writers who use it, so there is no need to divide it further. What varies is only who does the healing (Jesus, the disciples, even, mockingly, a hoped-for self-cure in Luke 4:23) and what is healed, not what the word itself means.

Underneath the medical meaning lies an older and broader one: to wait on someone, to render service, to attend to a person or a god. A doctor heals precisely by attending to a patient, which is how the one word came to cover both ideas, and in everyday Greek outside the Bible this serving sense was common.

In the New Testament this older meaning surfaces only once, in Paul's speech at Athens, where he says God is not served by human hands as though he needed anything. The picture is of priests waiting on a god, bringing it food and tending its needs, and Paul's point is that the true God needs no such attendance. Because this meaning appears just once in the New Testament and the medical meaning everywhere else, a reader meeting the word will almost always be looking at healing, not service.