From a jazz perspective, whenever a chord symbol contains an upper extension like 9, 11, or 13, the 3rd and 7th are implied. Moreover, in jazz, the chord C7 has many well-known voicings, including Bb-D-E-A and Bb-D-E-G. So if a lead sheet goes to the trouble of writing C13 when C7 would suffice, then this usually implies that the 13th carries special significance: the 13th should be included in the voicing or should be used for voice leading--or maybe the 13th appears in the melody. But C13 still contains an E and Bb, and it can also contain the 9th.
So a voicing like C11 implies use of E, Bb, and F--though not necessarily in that order, of course. But jazz songs don't often contain chords like C11 or CMaj11, because the E and F conflict, just as you describe. And if a pianist does see C11 on a lead sheet, then her eyebrows will raise and she will try to figure out what's intended. Perhaps the melody in that measure contains both an F and E, and C11 is supposed to convey this, and the pianist should follow the melody when voicing the chord. Or perhaps she is actually supposed to play a bona fide C11 voicing like Bb-E-F-A, which can work really nicely. (That particular voicing creates a cool opportunity to emphasize either the E or the F, which can change the sound.) Another good voicing for C11 is F-Bb-D-E.
In jazz, C7sus implies a range of voicings--as does virtually every chord symbol. C7sus could be played as F-A-Bb-D, G-Bb-D-F, Bb-C-D-F-A, D-F-Bb-C, C-F-G, C-D-G, Bb-D-F, etc. The main idea behind C7sus is that it does not contain the 3rd. One might wonder if this is sufficient to qualify as a "sustained" chord. But even if C7sus never resolves to C7 (with the 3rd), the ear still wants/expects to hear that resolution. In jazz, that implied/desired resolution--which is achieved by omitting the 3rd--is sufficient to qualify as a 'sus' chord.
Overall, C11 implies that the third is included, whereas C7sus implies that the third is omitted. However--and this might be where it gets weird--it is not required for every single C11 voicing to include the 3rd (or even the 11th for that matter). In a single jazz song, the band might cycle through the form 10-20 times. We cannot possibly constrain each voicing to the same 4 or 5 notes on every pass through the form, because the music will quickly begin to sound repetitive and stale. Improvised melodies often develop harmonically, starting with melodic lines that emphasize/end on the 1st, 3rd, or 5th and evolving toward melodic lines that more heavily emphasize alterations/upper extensions. The chordal instruments often do something similar. That sort of flexibility in the voicing is often essential to jazz--especially post-swing era jazz / bebop and beyond.