Was John Hancock really the first and third president of the US?

No, he was the 2nd President of the Second Continental Congress and the 7th President of the United States in Congress Assembled. This takes nothing away from him as a founding father, his importance in the Revolutionary Period of the US will always exist, both for his participation and for his leadership.

Yes, John Hancock was the first US President. The USA was born on July 4th, 1776 not April 30, 1789 when George Washington was inaugurated. John Hancock signed his name large on the Declaration of Independence not because he was making fun of King George's eyesight, but because he was the leader of the newly declared (u)nited States of America.