That's an interesting question. For example, if you're calculating the gravitational pull between the earth and sun F = G(m_1 * m_2) / r^2, you will reach for the distance between the centers of the two masses. This is the definition I intended and if you took a poll I think this is how most people would interpret it. I've never heard anyone interpret distance between two celestial bodies as minimum distance, although that's certainly a valid possible interpretation!
If you are doing something like shining a laser beam at the moon, or radar at a planet, then the distances are between the surfaces of the bodies rather than between the gravity centere. And if you are flying a probe towards the sun, the altitude above the surface becomes the more important number than the AU distance.
