If you want to get all definitional, an AU isnt the distance between the sun and earth. It is 1/2 the diameter of the earth's orbit. It is from one theoretical point to another and doesnt take the radius of either the sun or the earth into account. The real "distance between the sun and the earth" is 1AU minus the radii of the earth and sun. And comming up with a size/radius for the sun is itself a huge discussion.
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.
