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This is mostly about mechanical changes such as:

- Move InputMethodController-related logic inside InputMethodController
  and call it first - such that we can expand the logic there.
- Rename method name for better readability.

Also, there are subtle logical changes:

1) Try finishComposingText() even when the current input type is NONE.
   This should not have any effect.

2) When finishComposingText() returns false, we no longer call
   UpdateTextInputState(). This should not have any effect.

3) When finishComposingText() returns false, we no long call
   UpdateCompositionInfo(). This is used by Android and Mac.
   Previously, there was a chance that we propagate InvalidRange() all the
   way up to the keyboard app. Now ShouldUpdateCompositionInfo() returns
   false to make it clear.

This is originally found in resolving crbug.com/664317, but is also
needed for crbug.com/630137 because if we no longer cancel composition
on selection change, then WebViewTest may fail because it uses
TestWebViewClient which does not implement resetInputMethod().

BUG=664317, 630137

Review-Url: https://codereview.chromium.org/2561543003
Cr-Commit-Position: refs/heads/master@{#437854}