The convention with chords - any chords - is that they contain specific notes. Thus C major will contain C E and G. The order - and quantity - of each is not specified within 'C', so numerous different voicings (note orders) can be and are used. 'All the right notes, but not necessarily in the right order,' as dear old Eric once said ! Sometimes, even the 5 (G here) can be omitted. Main reason being it's already audible as an overtone of the root C.
Sometimes the 3 (E here) gets substituted for either a 2 or a 4, and the chord then gets titled Csus2 or Csus4. But in this particular case, the 3 stays, and a 2 is played as well. So, it's now Cadd2, although with the 2 (D) near the top, I'd argue that it would be more clear to call it Cadd9. It makes sense that if the 2 is in amongst the other chord notes lower down, that it would then be Cadd2. Subtle difference, but helping when the player reads the chart.