This is a great question, about a tricky distinction that throws many people off.
Centuries ago, chords were thought of as ONLY being made of stacks of thirds. ("Tertian" harmony). This meant that all chord tones could be thought of as odd numbers. (1, 3, 5, 7, and so on). To figure out a chord, you'd rearrange the notes until you got them into stacks of thirds, and then call the bottom chord the root. (An easy way to see "stacks of thirds" is to think, "every other letter.") For example, with your chord:
D C F A
you'd rearrange them until you got stacks of thirds:
D F A C
See the "every other letter"? That's D (E) F (G) A (B) C.
Here's one more, just to quickly illustrate: if you had the notes B C G E, you'd rearrange them into every-other-letter, which would give you a C major seventh chord: C (D) E (F) G (A) B.
This always works just fine as long as you've got four notes that are a third apart in some sort of arrangement, AND as long as you have a "rule" that chords can only be thought of as being made of ODD numbers, in stacks of thirds. According to this way of thinking, even what we think of today as an F6 chord in root position (FACD) would NOT be thought of as F6. It would be thought of as D minor seventh in first inversion. FACD couldn't be a root-position chord because C and D aren't "every-other-letter."
Fast-forward through time, though, and composers are constantly pushing the boundaries of harmony. What used to be "taboo" eventually becomes no big deal. Chords get more complex, and complex sounds begin to sound more "normal" to the average musical ear. This not only includes things like upper extensions (like C7b9) but it also includes the major 6th chord. This is a big deal because it represents a pretty fundamental difference. Composers began to *treat* a chord like FACD as if it were in root position (by emphasizing it on the I chord, using it as a point of resolution, etc.). This meant that we had a different "rule"! This chord, in its simplest, root position, was NOT made entirely of thirds, but instead it contained the 6th, an *even* number. The chord could be 1356.
So this is where we are now: the average ear tends to "work out" most typical tertian chords in the first way, above: whether we're conscious of it or not, on some level we mentally do the math and work out what the "real" bottom note (root) is, based on mentally rearranging into stacks of thirds. However, nowadays the major sixth shape is "allowed" into the mix, as a sort of exception. If we hear the chord FACD, we MIGHT hear it as F6, or we MIGHT hear it as Dmin7 in first inversion. Which one we hear has to do with how the chord is treated by the composer/player. Without surrounding chords to provide context, we tend to be influenced by the lowest note. With no other context, most people are likely to hear the chord FACD as F6, because F is on the bottom. Likewise, most people are likely to hear your original chord, DCFA, as a D minor seventh, because D is in the bass. It's possible for two people to hear a chord like FACD in two different ways and, in a sense, they could both be right. However, it's usually possible to go back and look at the recording/sheet music for context, and determine which chord the composer/player was probably intending for it to be.
Hope that makes sense! Great question!
James
