Let's start with the easy bits. The distance from the center of the Moon at the far point. The max distance that the Moon was from Earth was actually at the exit point. The exact distance involves some complex geometry. Another thing to keep in mind is the Earth and Moon are not exact spheres, the radius can vary quite a bit. The Earth's actual radius can vary between 6353-6384 km depending on exactly where you are, it is less towards the poles. The Moon's similarly can vary between 1736-1738 km. Thus whatever value we give could be off as much as 33 km, depending on which point of Earth/ the Moon we are talking about.

The distance should be (dMoon_Earth)-rE+rM+dA13. Let's use the values that will give the largest number for all of these, and we get 400,061 km, which is still less then the reported value. This would assume the min distance from the Equator of Earth (Unlikely) and the equator of the Moon (Likely).

Interestingly enough, the NASA article cites the Guinness Book of World Records as the source it uses. Other sources give even higher numbers, such as 401,056 km.

My guess is that they picked the distance to a point on the Earth, but not necessarily the closest point to Earth. A point on Earth that wasn't directly in the line of closest approach would give a slightly larger value, as is seen.