I think you are right. But I get the feeling that most tabs don't make that fine a distinction. So, I would just play that Cadd9 and think of it as a C2. But I am a real shortcut artist, and maybe I should be more exact.

Here is a C2 with no 3rd (no e). The difference in this chord and the one before, IMO, is pretty subtle.


E |-------------------------------|
B |-----------3-(d)---------------|
G |-----------5-(c)---------------|
D |-----------5-(g)---------------|
A |-----------3-(c)---------------|
E |-------------------------------|