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[Extensions Docs] Update API proposal process

Update the proposal process to use the new template. In particular:
- Update links to use the new template
- Remove sign-off labels from the bug, as they are in the template
  instead
- Slight rewordings and updates

Bug: None
Change-Id: Ifd9c59408396c8346e6d778839f9da7893a6bac0
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2028967Reviewed-by: Istiaque Ahmed <lazyboy@chromium.org>
Commit-Queue: Devlin <rdevlin.cronin@chromium.org>
Cr-Commit-Position: refs/heads/master@{#737006}