I just noticed after having submitted this that I have responded to a thread which was in most part from seven years ago. - Oh well- I'm sure the principle inquirer has long since settled on his options, but here's some ideas on the matter anyway, in case it's worth anyone's current consideration and to their interest.
It is often arranged and played in either version. It's a tune the associated traditions of which are very much informed by its having come from an era in which the transition from a modal to a tonal conception of music was still being worked out. I believe that our modern sense of tonality was directed towards the primary prevalence of major-minor modes, i.e., ionian and aeolian, the latter with its adjustments to fit, by their being the only two modes in which the tritone inherent to the diatonic system being resolvable to an interval supported by the tonic triad. Strict adherence to the notes provided by the scalar step profile of other modes introduces a sort of sonic "tension in the grain", which nowadays may have an exotic or archaic sort of appeal. But in those earlier times, tunes may have been notated and thought of as in another mode, such as dorian or mixolydian, but in practice reflecting the transition that was in the air, often were played with notes adjusted "out of place", e.g., in the case of dorian, with the sixth degree lowered, according to the musical direction of the phrase, or in mixolydian, with the seventh raised, at least in cadences. This can be seen in lute tablatures, in which what is notated is what one does to the instrument rather than what was the theoretical thinking with which composition was approached, and residual modal influence can be seen in scores as late as those by Bach, in which a dorian key signature may have been indicated, but the sixth degree is chromatically lowered every time it shows up.
I agree with those who suggest playing it how one likes to hear it. The lowered sixth (F natural) is like an inflection associated most directly with the fifth degree, the tone which it elaborates, and as such, represents the easiest melodic flow, the path of least resistance. The motion up and back to the true dorian sixth introduces a sort of "hitch" in the proceedings, which one may find to be intriguing. This is derived not only from the relatively increased whole step melodic excursion up from and immediately back to the fifth degree, but also from the note, in this exposed position, being a tritone away from the third, which is structurally prominent by being the note leapt to on the first strong beat of the melody, thus emphasizing that interval.
But the connotations of the choice of note here go beyond just the effect local to the measure. The B section of the tune features the chord on the natural seventh degree as "home plate", or, in the key you are discussing, a G chord, after having been hearing A minor. Some versions (again, in the key you discuss) begin the B section with a G harmonization, others approach it through a C. This sounds as temporary a shift in tonal center-- not strong or lasting enough to say a modulation has taken place, but something like. If in the A section, the sixth in its lower position has been used, this shifting of the ground on which one stands comes rather suddenly, and is a bit of a surprise. But if the sixth in its upper, dorian inflection has been used, and reiterated by the second time it comes around i the A section, then it functions like a far-reaching leading tone to the seventh degree, This gives the "sonic tension" of the upperly inflected modal sixth a direction, and resolution by which it presages the arrival of that chord. In this way, that tension is functional as a factor by which the A and B sections of the tune are organically tied together.
So-- play it as you like, but what is it that you do like? Ease of flow, and un-foreshadowed shift clearly dividing the two sections of the tune? Or, a promentory melodic "spike" in the A section which creates an incentive which, in it realization i the B section, lends a coherency to the piece as a whole? You decide.
This is not the only note which appears in variant form in different versions. You may also want to consider whether to use a leading tone consistently throughout, i.e., the # on G in arpeggiations at the end of phrases, or to reserve it only for final cadences.