Been out of Berklee a few years, but a few nuggets that might shed light:
In Jazz theory, the 7th scale degree is part of the chord. C7 isn't C (C+E+G) plus a B♭; C7 is its own flavor of C chord, C + E + G + B♭.
More generally, the ^3 and the ^7 are the two most important notes in the chord. The ^3 tells you its quality (minor, major) and the ^7 tells you its, uh, seventhness (dominant, diminished, maj7). It's common for players to leave off the 1 and the 5; they're low-information notes, and those fingers are better put to use playing the tensions, which are "chord extensions", the 9th, 11th and 13th. Critically, all three tensions are considered part of the chord, unlike Csus4, which means substitute a 4 for the 3, and don't play the 3.
C9, C11 and C13 can exist, but you won't often find C13 in particular like that. There's a whole set of memorizable rules to tell you which chord scales go with which chords performing which functions; if that C is a I chord in the measure, you're not likely to see a C11. You may see a C7(11), or C7(add11), which is IIRC the C chord plus an 11th note - but not the 9th. Then there's C7(#11), which you'll often find when C is the IV chord, 'cause the IV chord gets a Lydian scale.. but now I forget how that explains anything.
C+E+G+B♭+D+F+A can also be looked at as D-/C7, and often is. That whole "Upper Triad" polychord stuff is a whole semester to itself.