Yes, a sorcerer, but not a wizard, in the second option. The idea was to make wizard a subclass of sorcerer. Spellbooks allow greater spell access is the subclass thing. Which I feel works if you want wzards to have the gift for magic, like Harry Potter.
I said if the ability to use magic is a "gift" you are born with or gain from something, like the sorcerer, then wizards as sorcerers also have the "gift".
The second mind is a world where anyone can learn magic, you don't need it as a gift. If you think that way, it doesn't make sense that wizard should be a subclass of sorcerer. They have no "gift".
A wizard sorce of magic is learning magic -- nothing born with or gained by event. That is second option.