Great post Dusty, I hope everyone will read it because every sentence in your post contains clear and correct information that sheds light on this whole subject. 
I don't think I can answer the original question posed, but I am not sure there is a clear answer. You can tune a dulcimer however you want, but that doesn't mean that there is an official name for the tuning. The Canadian dulcimer player Rick Scott has written lots of tunes in strange tunings. He doesn't have names for them; he just tells you what notes he is tuned to.
There are basically two ways we name tunings. One is by the mode that is available on the melody string (assuming there are no half frets and the melody is played only on the melody string). In that nomenclature, we do not have a choice about the drone strings. They have to be the 1st and 5th notes of the scale, although they can be reversed. DAA is ionian and ADa would be a reverse ionian. DAC is aeolian and ADC would be reversed aeolian. What is reversed is simply the order of the drones. The tuning of the melody string has nothing to do with whether we call a tuning "reversed" or not.
The second way of naming a tuning is simply by labeling the strings according to their scale position. DAA is 155, as is GDD and so forth. DAC is 157, FAC is 135, etc.
(For what it's worth, I much prefer naming tunings according to scale position since it avoids the obfuscating modal names and is easier to transpose from one key to another. Additionally, since many of us have extra frets and fret across all the strings, we are not limited to a single mode in any given tuning, making the modal names technically incorrect and certainly misleading.)
So one way to label your tuning would be to establish the key and then determine the scale position of the strings. If @strumelia is correct, and you are playing Emma's Waltz in G aeolian (and the song is certainly in aeolian) then your Bb is the m3, the G is the 1, and the F is the 7. So we could say simply that you are tuned m3-1-7. But we cannot call that an aeolian tuning because the drones are wrong. I think perhaps the reason it sounds OK is that the minor third is one of the notes--and one of the defining notes--of the aeolian mode, so it sounds reasonable.
If you were to tune the bass string to D, so that you were in DGF, that would be a true reversed aeolian tuning.
You said: "B flat / G / F (melody string is F)"
In listening to your sound clip of your open strings, you have lowest string (bass) at Bb. The middle string is tuned to the G above the melody string's F. Thus, the highest sounding open string is the middle string G.
The youtube you copied below is playing Emma's waltz with a capo at fret one, resulting in the minor-sounding aeolian mode. This is what I guessed before with your tuning. On your open F melody string, the first fret of your melody string will be your 'home'/tonic note which would be a G (matching your open middle string in G). If you aren't using a capo (without telling us), you'd be playing the waltz in the G aeolian scale, in Aeolian mode which has a lonesome/sad feel. From the internet: "G Aeolian is the sixth mode of the Bb major scale · G Aeolian Scale Notes: G A Bb C D Eb F". Your tonic/home note is G, on the 1st fret of the melody string.
The reason your melody string is the F note LOWER than the middle string (usually it's higher than the middle string) is because if you tried to tune it to next f note one octave higher, at a 28"+ vsl that string would break no matter how thin a gauge you had. So you went with the F an octave lower which happened to be one note below the middle string G.
If you play a little snippet of an actual tune, a few measures, particularly the end... we could tell you what key and mode you are playing in, based on your tuning.
I am trying to understand your tuning. The melody string is tuned to F. The middle string is tuned to G, but is it higher or lower than the F. The bass string is tuned to B flat. Is this tuned lower than the F? I want to try tuning my dulcimer this way, but may need to change strings to do so.
Ken
"The dulcimer sings a sweet song."
Inversion of Gm7 according to https://www.scales-chords.com/chord/piano/Bbgf
What tune are you playing with this tuning? That will help us.
What fret does the tune end on? That will help too.
I'm 'suspecting' you are playing in G aeolian, with the tonic note being G at the first fret of the melody string, and the middle string being tuned to G the tonic. That would give you the minor/lonesome sound.
But tell us answers to my questions above.
I’ve a music question that I’m guessing someone here can answer. I stumbled upon a minor-ish dulcimer tuning that actually seems to work, in a haunting way, for a couple of minor-key tunes played noter-drone style. What the heck have I got here? In this particular case I’ve got: B flat / G / F (melody string is F). I don’t know if this is an inverted something or other, mode or what - but I’d like to know what to call this. For the moment, I’m immodestly calling it “Terry Tuning”. There’s no way that can stand! 🤓
I never had a music theory class so I suppose this could be an augmented or diminished 7th or something I’ve only ever seen in a guitar chord book. But I’m curious - it does have a haunting quality. A measured version of Emma’s Waltz is especially cool - to my ears late at night, anyway. The cats don’t seem to mind, at least, and my dog would hang close no matter what. Thanks to all. And as a friend says, Stay Positive and Test Negative!
