Two little letters are what causes the problem here. Take off the “er” from the end of “cooker” and the expression is fine. Leave it on however, and you’re talking about a machine, not a person.
So, just say, “I’m a good cook” and everything is fine. You could also say, “I’m good at cooking” or even add the type of food you are good at cooking, “I’m good at cooking kimchi fried rice.”
A side point here is that “making” is a more common verb than “cooking.” You can “make coffee,” “make popcorn,” “make kimbap,” or even “make ice cream,” but it would be odd to use “cook” with any of those foods.