Also, being in free-fall relative to the earth isn't locally distinguishable from being in free-fall relative to both the earth and the moon. Tidal effects are

Also, being in free-fall relative to the earth isn't locally distinguishable from being in free-fall relative to both the earth and the moon. Tidal effects are caused by the gradient of the gravitational field across significant distances. And the gradient drops off in proportion to the inverse cube of the distance, not the inverse square. Which is why tidal effects are only observable from the moon and sun, not from other celestial bodies. (Jupiter's gravitational influence on the Earth is about 1% that of the moon, but it's so far away that the tidal effects are infinitesimal.)