For tidally locked binary stars, the two points in question are known as the substellar points.
For a tidally locked exoplanet, the point closest to the star would also be known as the substellar point. If the star was also tidally locked to the planet, then there would be a subplanetary point.
For a moon locked to a planet, the point on the moon would also be the subplanetary point. If the planet was mutually locked, it would have a sublunar point (not sure whether this latter only applies to the Earth's moon).
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