Lots of songs use both natural 6 degree and b6 in the same piece. Originally the raised 6th in minor was to close the gap between the 6 & 7 scale degrees in harmonic minor at a cadence but since then it seems the natural 6th degree is used wherever or whenever other than at a cadence. Is this use considered dorian mode or just melodic minor used in a non-conventional way? Or does the 7th degree also have to be rasied to consider melodic minor?
There is no single coherent theory from which all words come from. Particularly in music and other areas of human culture. There are only different contexts, situations and perspectives in which people use words, in order to communicate something to someone. When interpreting words and sentences, you inevitably make assumptions about context, perspective and intentions. Sometimes your assumptions can be right.
If a person was thinking about the Dorian mode scale when making a decision to use a sixth, then the sixth can justifiably be said to have come from the Dorian mode.
If a person was thinking about the melodic minor scale when making a decision to use a sixth, then the sixth can justifiably be said to have come from the melodic minor scale.
If a piece of music, or at least a section of music that's being talked about, uses scale alterations, e.g. changing scale degrees to sharp/flat/natural, then in my opinion it would be misleading to call that section "modal". When talking about modes, for example Dorian, then I make two assumptions about the harmony at that point: (1) that there is a clear tonic note, and (2) the set of intervals around the tonic are fixed, and there are no chromatic alterations.
In pieces in minor keys, scale degrees are often altered along the course of the piece. For example, scale degrees are changed so that the set of notes can be seen to form the melodic minor scale, or the harmonic minor scale. Or the natural minor scale.
However, the melodic minor scale can also be used with rigid harmony, so that no scale alterations happen. In that case it is used modally, and it can also be called "Jazz minor", to separate the use from the other "melodic minor" concept, which might mean - at least for some people in at least esome contexts - the simultaneous possibility of different chromatic alterations of the 6th and 7th scale degrees, depending on what the melody and chords do. That is not "modal".
Can a piece of music be legitimately called "Dorian", if the seventh scale degree is never used, and for example only six out of the seven theoretically existing scale degrees are ever explicated? I think so. The sixth is a characteristic note that makes Dorian sound like Dorian.
This question cannot be answered definitively, because music is more complex than the categories it's pigeonholed into. It's quite possible to have a piece that utilizes the lowered and raised sixths, and the natural and raised sevenths, in such a way, as to defy classification.
But typically, dorian pieces have no lowered sixths or raised sevenths, and minor pieces (as Aaron said in his comment) almost always use the raised seventh. But there is music in between that blurs the "boundaries".
The sixth (and seventh) steps in minor are mutable. Both the upper and lower step can be seen as diatonic in a minor key. All combinations often occur in the same piece (except the raised 6 and lower 7 are rare as that would tend to sound like a major scale a whole step below the tonic or the Dorian mode on the tonic.)
I posted some typical mutatable note usages somewhere here (and on Quora) but I can't find them. (I even have them somewhere as text files but I don't know on which computer.)
Jazz and pop theorists do have a different viewpoint but I'm not familiar enough with them to give a discription.
There really isn't a "the sixth" in minor key music. Meaning it is not one, single pitch. The sixth scale degree varies.
I suppose you could say there is some historical basis to say it comes from dorian. The "dorian" key signature was used in the Baroque era. But that really doesn't say much. Despite the "dorian" key signature, the sixth degree was lowered by accidental throughout scores.
Beyond that this question has been asked and answered many times.
The sixth and seventh degrees in minor vary and are determined, on the whole, by harmony first and melodic motion second. Basically when the harmony is dominant, the raised sixth and seventh are used.
It occurs in both. So if you feel the need to justify using it, you've got two choices!
It's in the category, I think, of 'non-diatonic things that are so common that they're almost honorary diatonic'. The ♭3 and ♭7 notes in any melody influenced by the last century of African-American music. The minor IV chord used in a major key. The ♭VII chord. So mainstream as to require no 'borrowing' or other excuse.