I think everyone just defines it as a space that has an inner product. If it has another norm, just replace it with the one from the inner product. So, I don't know that there's any difference in definitions here. Having an inner product gives you a norm. It might be a different norm than some other norm that you might put on the space. So, the space is going to be an inner product space with respect to one norm, and it might not be with respect to a different norm.Ah, I see, some people define an inner-product space as one where the norm derives from the inner-product.
It's a strange statement to say that C[0,1] with the sup norm AND inner product given by the integral is not a Hilbert space. It IS a Hilbert space. It's just that it now has two different norms on it, instead of one. It's better to say that the sup norm doesn't come from an inner product. The space by itself without the inner product is something that gets turned into a Hilbert space or Banach space, maybe in more than one possible way. It's not Hilbert until AFTER you put the inner product on it. And it's not Banach until AFTER you put a norm on it. There could be more than one possible inner product or norm. Just as a set could become a group in many possible ways, unless it has a prime number of elements or something, so that we know that it's cyclic.