Is there a concept of information that distinguishes between "computed" and "non-computed" information?
The distinction between information theoretic and computational knowledge is an issue which is fairly common in cryptography.

Consider for example the one-time-pad encryption, that given a message $m\in\{0,1\}^l$, encrypts it using a secret key $k$ (chosen uniformly at random) by $m\oplus k$. Clearly, one learns no information from seeing the encryption of one message (it is a uniformly distributed string), so you could say that you have information theoretic security, which should make you very happy. However, if you insist on such a strict notion of security, it turns out you cant encrypt messages who are longer than the key, and if you want to do that, you should settle for a weaker notion, which is security against bounded resources adversaries (see semantic security).

One example which seems to fit your question well is one way permutations. A function $f:\{0,1\}^*\rightarrow \{0,1\}^*$ is called a one-way permutation if for every $n\in\mathbb{N}$, $f|_{\{0,1\}^n}$ is a permutation of $\{0,1\}^n$, and for some bounded resources adversaries, almost no knowledge of $x$ is exposed from $f(x)$. Since $f$ is a permutation, $f(x)$ completely determines $x$, but it is extremely hard to recover.

That said, I will strengthen Raphel's comment and say that there isn't a general definition for information (or what you call computational information), this is an issue which strongly depends on the context.