It’s interesting to compare (add9) to (add2) or (sus2) chords. There might be a significant difference. I’ll explain this with reference to open position guitar chords. Look at a regular D chord. If you open up the high E string you got a D(sus2). The “sus” implies that the third degree (F#, normally on the first string) is not played and the (2, E) is substituted for the third. Now consider the same D chord with the E note on the second fret of the fourth string. From the highest string down, the notes are F#, D, A, E. This would be an (add2) because the complete major triad is still being played.
Now lets consider a C major chord in first position. If you add the D note on the second string, third fret, I’d call this a C(add9) because again, you have the full C triad sounding on the 5th, 4th and 3rd strings. If you open up the fourth string of a regular C chord it’s more accurately a C(sus2) or C(add2) depending upon whether and E notes are sounded.
Oh. And if you add the D note to a C7 chord it’s definitely a C9 as chords named with numbers higher that 7 are implied 7th chords with compound intervals added to them.
Having fun? Let’s go back to the D(add2) played with E on the fourth string. You could also call this an E11 or an A6(sus4). Any chord with four or more different notes has at least two different root names.
I could go on, but I better stop.