Words to mean one taking only part of the other's words?

Solution 1:

Assuming I'm understanding the situation correctly, Person A would probably suggest that Person B took his statements out of context. That is, while B's quote might have been accurate on a word-for-word basis, he left out the surrounding explanation changing the overall meaning.

Solution 2:

To add to @Dusty a word for "taking only part of another's words" is misquote

Another phrase that come to mind are "twisting my words" or in your context "twisting A's words". This phrase implies there has been some malicious intention on B's part and B is purposely bad-mouthing A or fudging A's words.

twist someone's words (around)

to restate someone's words inaccurately; to misrepresent what someone has said.

Stop twisting my words around! Listen to what I am telling you! You are twisting my words again. That is not what I said!

Solution 3:

If Person B has not bothered to listen to all of what was said to them in the first place, (i.e., if they tuned out after listening only the first part of what Person A had to say), you could describe Person B as exhibiting selective hearing.

Solution 4:

Applicable in your situation and also a common practice in negotiation, discussion etc is ... oversimplification and misrepresentation.

• B is oversimplifying A's reasoned position.

• B's restitution of A's position is oversimplified.

  oversimplify(v) To simplify to the point of causing misrepresentation, misconception, or error.

  misrepresent (v) To give an incorrect or misleading representation of.