Hi. If I knew exactly what was going on, the checkbox probably wouldn't be there in the first place.
In theory, the neutral axis (the plane of wire where there is no compression or stretching when bending) should be the ideal diameter to use. At least that is what I think/feel. And a simple way to get pretty close to the neutral axis in a symmetrical wire is by adding half of the wire thickness to the inner diameter.
In practice, using the default setting (outer diameter), usually seems to yield the most accurate result. What I mean by that is the result that matches the measured resistance more closely.
Maybe this is because the wire stretches more than it compresses, so the resistance increases compared to a non-wrapped wire. Or maybe the deformation of the wire leads to a net reduction in cross section area. Maybe metal fatigue is increasing the resistance. Maybe there are other factors at play as well or instead. I simply don't know.
So anyway, experience suggest that in everyday use, the default setting (outer diameter) will usually give you the most accurate result. Except when it doesn't. I'm sorry that I cannot give you a better answer, but the coil is an analog circuit, and there are so many factors at play that nobody (except possibly Stephen Hawking) can keep track of them all.
If you click the "How it works" link in the bottom left corner of the calculator page, I have mentioned some of the error sources that I've been able to think of.