Remove reference to private web/ API from Tab public API.

Move references to private web/ API from Tab public API to the
private API. The goal is to prevent adding new uses of those
methods in order to simplify their eventual removal.

Add missing includes to fix the build.

Bug: 620465
Change-Id: Ib30593b674e43a9eb17312db17bfd3436a6378ba
Reviewed-on: https://chromium-review.googlesource.com/635566
Reviewed-by: Eugene But <eugenebut@chromium.org>
Commit-Queue: Sylvain Defresne <sdefresne@chromium.org>
Cr-Commit-Position: refs/heads/master@{#497415}
13 files changed