A C major triad contains the notes C, E, and G; add a Bb to get a C7 chord, which is often called a dominant 7th chord. Extended chords are usually thought of as extending a 7th chord.
Now, a C9 chord is understood to be a C7 chord with an added D, or 9th. To form a C11 chord, add an F (the 11th) to a C9; to form a C13 add an A (the 13th) to a C11. So, a complete C13 chord contains the notes C, E, G, Bb, D, F, and A (the 1, 3, 5, b7, 9, 11, and 13).
Generally speaking, it is considered fine to omit the 5th of any chord with a perfect 5th (diminished and augmented triads should probably keep their diminished or augmented 5ths, though). As a rule of thumb it is fine to omit the lower extensions, too; so a C11 might omit the 9th, and a C13 may omit either or both of the 9th and 11th. Following a comment from @user1079505, the 11th is usually omitted from major chords in any case because the 11th clashes with the major 3rd. All of this is subject to the conditions of an actual musical context.
In addition, the guitar has its own constraints. It can be difficult to play some chords in 6-note voicings, or it may be difficult or impossible to find a playable voicing that sounds good. Guitarists routinely abbreviate chords because the abbreviated chords are easier to play, and often sound better than their more fully-voiced cousins. Playing with other musicians also provides plenty of reasons for leaving notes out of your chord voicings to leave space for other players, but that is not what is at work here.
Here is one extremely common voicing for C13. Note that I am showing two "grips", i.e., two fingerings, for the same voicing (arrangment of notes):
This voicing is C, Bb, E, A. Both of these voicings have the root first, then the 7th, then the 3rd of the chord. This configuration of 1-7-3 is called a shell voicing, and it contains the essential notes to communicate 7th chord quality: the root, the third (is it major or minor?), and the seventh (is it a major 7th chord, a dominant 7th chord, or a diminished 7th chord?). After getting those three notes down, you can plunk down a 13th and you have the skeleton of a 13th chord.
Or you can try adding more extensions, and here is another version that manages to fit both a 9th and a 13th on top of the shell voicing:
This voicing is C, E, Bb, D, A, and it is the voicing described in the OP post. The first three notes of this voicing also form a shell voicing, this time 1-3-7. Note that this 13th chord is related to what is probably the most common voicing for a 9th chord on guitar:
The first one is usually seen with the 5th (the G on top) omitted because otherwise it is a bit of a finger-squisher (and it doesn't really need the 5th); the second chord shape is seen all the time, and it is not uncommon for the 5th to be omitted on this one, too. This second shape for C9 can be turned into a C13 by placing the pinky on the 5th fret (A), replacing the G on the 3rd fret, i.e., by replacing the 5th of the chord with the 13th.
Any musician might abbreviate chords for a variety of reasons, and there is nothing wrong with that; unless one is being pedantic or very particular about a specific voicing, one would still call a C13 which was missing the notes as described above a C13 and be done with it. There are a lot of circumstances when even the root of the chord is omitted, and it is still called a C13.
Usually when a composer or transcriber wants particular voicings, they write them explicitly on the staff. Coming across a mouthfull like C9(omit 5, add13) is rough when you read a chart, where C13 is a lot easier to digest. Chord symbols are really just indicators of the prevailing harmony in a piece of music. Interpret them under the constraints of your instrument, abilities, and musical context, and everything will be fine.