What determines what we call a chord?

This one's easy: It's D9. D doesn't belong to an F triad... and your chord doesn't have an F triad either - the note on the 2nd fret of the 6th string is F#.

The notes you have are: F#-A-D-A-C-E

You can start with the assumption that any note can be the root. Although a root doesn't have to be present in some voicings, it's there pretty often, and almost always in open position chords. So this chord could have a name that starts with F#, A, D, C, or E.

Let's lay out each of those major scales, and see where the notes fall:

F#-G#-A#-B-C#-D#-E#-F#
A-B-C#-D-E-F#-G#-A
D-E-F#-G-A-B-C#-D
C-D-E-F-G-A-B-C
E-F#-G#-A-B-C#-D#-E

Now the chord tones; I'll do them in the order they appear on the strings, F#-A-D-C-E:

F# root = 1-b3-#5/b6-#4/b5-b7
A root = 6-1-4-b3-5
D root = 3-5-1-b7-2
C root = #4/b5-6-2-1-3
E root = 2-4-b7-b6-1

When we build chords, they're done in thirds. You really want to see only odd numbers - 2s become 9s, 4s become 11s, and 6s become 13s. So I'll re-write these in numerical order this time:

F# root = 1-b3-b5-b7-b11
A root = 1-b3-5-11-13
D root = 1-3-5-b7-9
C root = 1-3-b5-9-13
E root = 1-b7-9-11-b13

Each of those could actually have a chord name.

F#m7b5/b11
Am6/11 (the 13th is called the 6th when there's no 7th in a chord)
D9
C6b5/9
Esus11+5 (the 11th is called the 4th when there's no 3rd in a chord)

Looking over those names, the D9 is pretty obvious - the simplest name is usually the right one.

...and don't forget context.

yeah, there's context. I'd actually typed all that out too, but figured the post was too long already :)

Sometimes you'll have notes like A-C-E-G that can be considered Am7 or C6 (C-E-G-A). The names are synonyms of each other - they'll sound exactly the same, but only one name will be right in context.

If you had the progression:

Bm7 - E7 - ?

using Am7 makes sense - the cadence is V-i in A minor. The name C6 wouldn't work in that context. On the other hand, if the progression was:

F - Em - Dm - ?

the C6 would make sense, and the Am7 wouldn't.

Chords move logically, usually by fourth or fifth (as in the first example) or by step (as in the second example). When you can legitimately call a chord by two different names, it's context in the progression that determines which one is right.

Other keys to look at (if you have a full arrangement) are the vocal line (or the melody line on a lead sheet) and the bass line.

Very often the bass will be hitting the root of the chord, and the melody line will often give you a hint of what key the chord should fit into.

STOP!!! Your making my head hurt. Can't we just play and have fun?

You call the triad what you want, for example

A - C - E is a minor

E - C - A I a minor 6
4

or second inversion

how ever that triad could also be interpeted as any type of chord you want.

Containment team: We have a theory leak in the beginners section... we're going to need this cleaned up... stat.

Well done Tom.