Your problem is not that you need to persuade your GM to change the way the Warlock class works; His problem is that he has fallen prey to what I am going to start calling "the Bugbear nomenclature.""
Each character class is a collection of skills built around a concept that is frequently found in works of fantasy fiction. These classes have names, presumably because it becomes awkward to say "I am playing a person who uses really competent weapons and armor!" all the time. Therefore, the fighter is someone who can use armor and weapons well. The wizard class is someone who uses magic and keeps spells in books. You will have solved this a few minutes after reading the class descriptions.
However, being a member of a class does not mean that your character identifies with the Name of that class: a fighter could also be a Viking assailant, a wandering knight, a conqueror or any of the thousands of other possibilities, and similarly, there are many characters in fiction who cast spells and maintain a library, and most of them are not called "magician". The Bugbear nomenclature arises when we forget that class names are only a convenience of game jargon and we begin to think that they are recognized in the universe.
To be fair, this is an easy trap to fall, since often the words used for class names do they exist in the universe: they were taken from the fiction that inspired the game, and most campaign settings are also inspired by that same fiction; So, most D&D settings do they have people called magicians, and most of them They are members of the assistant class, because the assistant class is good enough to represent which attendees are in that environment. You just need to remember that in many of these environments, the nomenclature in the universe does not necessarily align perfectly with the terminology of the game; In most environments, you can present yourself as a "thief" and people will not necessarily assume that your character is a rogue with the archetype of the thief, since the only qualification on the stage to be a thief is to steal things and any class can do that.
Returning to his example: You want to make a character that is a sorcerer in the campaign universe; a person who uses subversive magic as an alternative to physical power and social skill. You want this to be backed by the mechanics of the game and its GM. That is all reasonable. Your goal is twofold: You want to find a class that suits your concept, and you want to work with your GM to have a shared understanding of what your witch pretends to be.
As your GM has pointed out, the sorcerer class is not really what you are looking for. But its concept is still quite broad, and that means it is flexible; With the permission of your GM, you can choose any class that fits your concept, be it magician, cleric or, well, anything that uses magic, and that your character calls herself "witch" in the conversation.
Once you've chosen a class whose skill list sounds like your vision of a witch, explain your idea to your GM. It seems that your GM is not opposed to the concept of your character, so I hope he is receptive. As long as he agrees that there is no great dissonance between what your character can do and what she calls herself, you will find that this solves your problem. You might have a little round trip if you have existing nomenclature plans for magic users in the environment, but eventually you will have a character that you are happy with and that your GM understands well. It is win-win!