I think of a 4 as a 13th.In this context: C Major 7 with sharpened 4.
You could use "add". Some people think that's clearer. It's the same meaning but just a matter of preference.
You'll find in jazz (and pop) that sometimes things are not "properly" designated. There is some flexibility in notation, usually the default is what works and what communicates the best. Don't get too black and white on this.
If you use "aug" instead of "+" you run the risk of confusing the designation pertaining to the basic chord and not the extension. Would you notate a chord C Maj 7, Aug 4th? I would default to other notations to avoid ambiguity.The proper terms are 'augmented' for raised pitch and 'diminished' for lowered so they just use the 'plus' and 'minus' signs.
This one is very confusing, I would look at that quickly and think C E G# Bb DC7+9 would be C E G Bb D#