Revert "Update typedef syntax to use Function notation and turn on lint for old notation. (#18035)" (#18041)

This reverts commit 32586020.

Fixes #18028

Just changes typedef declarations, no logic changes.

This updates AppContext per the recommendations in #15352

Fixes #15352

This class lives in the Context and allows callers to "inject"
flag values, where flag values are first extracted from the
command arguments, then from the global arguments as a fallback.