First: awesome process, @Jazzbass19!
The two quotes you used (from @Gio and myself) say exactly the same, albeit using different nomenclature.
Your charts are spot-on and you got almost all the music theory right (as far as I can tell). However, when you say “you need to flat the C# to a C”, you sort of say that you adjust the scale to conform with the funk box. That is not “allowed”. What I mean is you can’t just flat the seventh in a major scale, because then we no longer call it the major scale, but it is now a different scale (it is, in fact, the myxolydian scale).
BUT, as long as the chord (played by the piano or guitar) you are dealing with is a simple D major chord, it only consists of the triad (root-major third-fifth, i.e., D-F#-A). In that case, you can add the flat seventh from the funk box without any problem! If, however, there was a D major 7 chord, which is composed of the triad + the major seventh (i.e., D-F#-A-C#), then, of course, you can’t use the C.
So, as you have concluded, the note selection of the “funk box” is kind of like a “one size fits (almost) all”, as they can be played along (in your example) a D major chord, a D minor chord, a D minor 7 chord, and a D dominant 7 chord (which is D-F#-A-C), but NOT with a D major 7 chord!
The reason why the key is given as D minor is that while you can “get away” with playing the notes of the funk box here, sometimes you might want to use other notes as well, and then you need to know the key more specifically, in order to specifically avoid the major third or the major sixth (here: F# and B) and only use notes from the minor scale.
Finally, the “box” metaphor has nothing really to do with the underlying harmonic theory, but is a visual aid and stems from the fact how the bass works (mainly, the A string is a fourth up from the E string, the D string is a fourth up from the A string etc). It is easiest to memorize this as a box starting with the root, and the notes in the box can be conveniently fretted without much hand movement, but the notes you identified (D, A, C, D) can be found again and again all over your fretboard, and they are all equally valid.
I hope this is not more confusing that it is helpful! In any case, please keep asking