Factory pattern implementation variations

I see many people using something like this and call it Factory pattern.

class Factory {
    public IProduct Create (ProductEnum type) {
        switch (type) {
            case ProductAType:
                return new ProductA();
            case ProductBType:
                return new ProductB();
        }
    }
}

But isn't the real Factory pattern the one where you have one factory for each product type? So when you create new implementation of IProduct you have to create a new factory class and redefine the Create method. Something like in this picture:

Is the first example also Factory pattern or it is something else, and should that be used or not?