the face, and by extension the presence a face conveys, the outward look of a thing, or the person the face stands for
The basic picture is the front of the head, the face that another person sees and meets. From that one concrete image the word reaches out in three directions that the New Testament uses constantly.
Because you meet someone by facing them, the face comes to mean presence: to be 'before someone's face' is to be in front of them or sent ahead of them, to be "face to face" is to be together in person, and to flee 'from someone's face' is to get away from them. Because the face is the side of a thing that shows, it can mean the surface or outward look: the face of the ground, the face of the sky, or how a person appears as against what is in the heart. And because a face identifies who someone is, it can stand for the person, especially in the old expression "to receive the face," which means to take someone's side because of who they are rather than what is right.
These are not separate words that happen to look alike. They are one word stretched along a single line of thought, from the literal face outward to presence, appearance, and person. Much of this range is inherited: the Greek writers are leaning on a Hebrew way of speaking in which "face" already carried all of these uses.