13.4Cadd9 is not the same as Cmaj9

This distinction matters.

Cadd9

C major triad:

C E G

plus D:

C E G D

There is no 7th.

Cmaj9

Cmaj7:

C E G B

plus D:

C E G B D

So:

Cadd9 = 1 3 5 9

Cmaj9 = 1 3 5 7 9

The 7th is the difference.