I believe it has a few names, I call it a Suggestive chord.
The practice is especially prominent in guitar playing, because the wide spacing and only having 6 strings limits the number of tones that can be played at the same time.
suppose you want to play a C13, this consists scale wise of 1 3 5 b7 9 11 and 13. with 6 strings and 4 fingers on your left hand, playing the chord with all notes at once is impossible.
On piano this limitation is less so, which brings me to...
From a sonic point of view though, playing 7 tones creates an extremely dense sound. as far as intervals go you have 5040 different combinations happening! (7x6x5x4x3x2).
Depending on the mood of the piece or what you're going for often a simpler chord which includes the 13 is more (subjectively)appropriate. For example if an add 13 chord is used instead, you only have 4 notes happening at once which is a less dense 24 intervals at once. Then of course you have all the choices of which notes to include, the voicing and all that other fun stuff reserved for somewhere else.
For guitarists reading this who want to learn more, you could do worse than Ted Greene's Chord Chemistry
I'm sure a piano player could reccommend a similar book that can be mentioned here, and purchased by me ;)
Does a chord need to include its root?