Agreed, I think your points get at the right point. I would go further and say - we consider meaningful "intelligence" as having to do with "consciousness", and its absense causes the disconnect between Wolfram's definition and our intuition about human intelligence.

Defining consciousness is its own challenge, but one definition that I like (as a layperson) is that it consists of an approximate model of our self, that emerged as the brain recursively modeled itself while attempting to model other's behaviours, a critical trait once group co-operation and adversity became key to our survival.

This seems correct to me, a hurricane exhibits computational complexity, but it does not have a sense of self, and does not use that sense of self to make decisions.

Of course, that sense of self can also be broken down into manual computation, it's just a very complicated computation. For example, a bayesian interpretation of that "approximate model of our self" might define it as a generative model or joint distribution of actions and states, embedded within a markov decision process (MDP). Modeling, or speculating about your self would therefore consist of sequentially sampling from the joint distribution over several timesteps of the MDP.