Does precession affect this measurement? I don't have the astronomy vocabulary to ask this question.   You can find a better picture at https://www.timeanddate.com/astronomy/axial-tilt-obliquity.html.
I want to know if the distance between points A &amp; B on my graphic has a name, and if it would grow shorter or longer as the earth's rotation slows down.  Would this distance only reflect the 23° wobble in the rotation of earth or would it also reflect Precession of the Equinoxes.
No special name.
The problem that I see here is that the distance will vary over the year.  At the moment of solstice, you can easily find the distance by basic trigonometry:

If the center of the Earth is C, then $AC/AB = \tan(23.5^\circ)$ so $AB = 6370/\tan(23.5^\circ)=13000\, \mathrm{km}$

At any other time the distance will be different, and at the equinox the axis of the earth is parallel to the tangent plane of the Earth, so the distance is infinite. So the distance AB is not a fixed value, and varies over the year.
In the longer term the precession of the equinoxes will affect when solstice occurs, but not the angle 23.5.  This angle also varies in the long term between 22.1° and 24.5°, over a cycle of about 41,000 years.