What is the most common term for...

[Tristana]

...when responding to the argument, instead of responding to the big picture, one takes a snip of that argument and responds to that incomplete though and zooms way too far into the word-by-word rather than the sentence of the whole?

Sorry, this isn't necessarily related to leftist politics, but related more towards argumentative discourse (which I guess applies indirectly)

[The Idler]

http://rationalwiki.org/wiki/Cherry_picking

[Comrade #138672]

It could also be a strawman.

[Comrade #138672]

Because cutting in the argument like that only serves to misrepresent the argument.

[Clarion]

Rigour.

If one clause of the argument fails then the who argument fails.

[Deity]

Rigour.

If one clause of the argument fails then the who argument fails.
Not at all. Taking a couple words or sentences out of context and arguing against the incomplete thought does not make the original argument a failure.

[homegrown terror]

nitpicking.

[conmharáin]

It's like when you create some sort of parable to illustrate your point and your opponent tackles the details of the parable rather than the substance of the idea.