to greet: to acknowledge another person warmly with a word or gesture of welcome, whether in person or by passing word along
At its heart this is the ordinary act of greeting. It covers everything a friendly hello does in any culture: stopping to acknowledge someone you meet, sending your regards through a third party, saying goodbye, paying a respectful call on someone important, or hailing a person with honor. The single thread running through all of these is the warm recognition of another person, offered openly rather than withheld.
The shades come from the setting, not from a change in the core idea. When two people meet, it is a face to face hello (Mary greeting Elizabeth). At the close of a letter it becomes the standard way to send regards, which is why Paul's letters end with long strings of it ("Greet so and so", "so and so greets you"). When someone is leaving, the same act becomes a farewell. When the person greeted holds high office, the greeting carries the weight of paying respects. When it is loud and public, it becomes a hail or acclamation.
Because greeting was a social marker of who belonged to your circle, the word can also carry quiet force. Jesus points out that greeting only your own people proves nothing, since even outsiders do that much. Sending the missionaries out to "greet no one on the road" was not rudeness but a way of saying the errand was too urgent for the long, formal exchanges a roadside greeting required.
The most striking stretch of the word is figurative: the people of faith are said to have greeted God's promises from a distance, like travelers catching sight of a homeland they have not yet reached and waving to it across the gap.