The DOS point is correct in spirit. Google however did not in any regard start as a feature of Yahoo, it was an entirely stand-alone search engine for nearly two years prior to the Yahoo deal. Just one year after launching publicly it was handling several million searches per day directly on their own site. Google had already become very famous, with people going directly to Google's site to use their search engine before Yahoo signed a deal to use them to power their portal search. Just prior to the Yahoo deal, Google was routinely handling 15+ million seaches per day - a large figure at the time.

Here is a Salon article from December 1998 noting Google's superiority and promoting google.com specifically:

https://www.salon.com/1998/12/21/straight_44/