a chord is a key-combination like Ctrl-w—it is so-called because it is like the chords a pianist plays, with many keys pressed
iw is really more of a key sequence.
/end pendantry :)
Your visual-mode mappings are screwing with the usual iw sequence.
Once started with v, Visual mode does what you tell it to. In this case, you've sent ...
It'll be hard to get manually entered keys much shorter than you guys got it in comments. But if you use a mapping you can repeat the "jump to quotes and enter insert" action easily...
nnoremap XX 2f"ci"
(Replace XX with desired key combo.)
Then you just do XX, type in some text, <esc>XX, type in some text, <esc>XX, ... until you've changed ...