I think "quoting out of context" exactly describes what you are referring to.
If I quote you accurately and fairly, it is still very likely true that reading the full context from which the quote was taken would give additional information, nuance, etc. But any excerpt is, by definition, not the complete text, and unless the rest of the text was meaningless jabber, presumably it adds additional information or clarification.
Like many idioms, the phrase "out of context" has a more specific meaning than the sum of its parts. Any quote that does not include everything that the person ever said is "out of context" in the literal sense of the individual words. But that's not what we mean when we say that. We use the phrase to mean that by failing to include larger context in the quote, you have altered the meaning of the words.
I saw a news story a couple of years ago where a political candidate made a campaign commercial that included a video clip of his opponent saying -- and I don't want to get into the specific politics and distract from the point, so let me just say that candidate A showed a clip of candidate B saying "X". A reporter interviewed A, and showed a slightly longer version of the lecture that that clip was taken from, in which we saw that what B actually said was, "Don't say, X." So A clipped off the "don't say" to make it look like B was saying exactly the opposite of what he actually said. The reporter (rather rhetorically) asked whether this was not taking B out of context. A defended himself by saying that the reporter was taking A out of context by concentrating on this one campaign ad.
I think this pretty neatly illustrates the point. A took B "out of context" by clipping off the rather critical words "don't say" from the qoute. But the reporter was not taking A "out of context" by failing to mention everything else that A had ever said in his campaign. The issue about B was whether he was for or against X. By chopping out the "don't say", A was spreading a lie. The issue about A was not what his positions were on a hundred other issues: it was whether or not he lied about B. Even if A showed that he had criticized B 100 other times with statements that were absolutely fair and accurate, that would not change the fact that at least this one time he lied. Unless A could show that by taking some larger context, you would see that in fact he did NOT misrepresent B's position on this issue, then he was not quoted out of context.