The out of the box Traffic Sources report (Referring domain, Search Engine, etc) are based on the Internal URL Filter that you have setup. Google will come under referring domain, as it is a referring domain to your site.
The Marketing Channel reports are separate from the above reports, and are highly customizable, as you have shared the screenshots of the rules. To classify the traffic correctly, (1) review your conditions, and (2) review your order of conditions. If this is properly setup, your marketing channel should be fine.
For correcting your internal domains, I would suggest bringing the internal channel condition above referring domain condition.