A protocol (and in general, a cryptographic construction) satisfies information-theoretic security if no adversary can break the system, no matter how powerful the adversary is. The term "information-theoretic" is rooted in the idea that the leakage from the interaction can be studied from the perspective of information theory, and it can be concluded using these tools, which typically involve a simple of mix of statistics and probability theory, that the interaction with the system really leaks almost nothing. Now, perfect and statistical security are two forms of information-theoretic security: in the former, there is zero leakage, but in the latter there is a negligible leakage than can be made smaller and smaller by appropriately choosing a statistical security parameter.
Now, a protocol, or again, in general, a cryptographic construction, can satisfy computational security, meaning that it is only secure as long as the adversary has bounded computational resources. This is typically the case when tools like encryption (usually with some kind of homomorphism) are used, which have an associated computational problem that the adversary should not be able to solve in order to guarantee security of the system. The way security of these tools is formalized is by means of a computational security parameter, which, as it grows, it makes the underlying computational problem harder to solve, hence providing much more confidence (but also, typically, making parameters worse).
Although researchers can be a bit lax with language, the most typical situation is that you will find the type of security explicitly stated, e.g. ...protocol X instantiates functionality Y with perfect security/statistical security/computational security... However, sometimes this is assumed to be known from context, or the focus in terms of clarity lies in other aspects of the construction, so the authors will not state this quite explicitly. As a rule of thumb, if there is any construction that assumes a bounded adversary, which typically include encryption, signatures, commitments, among others, then security is for sure computational. It should also be mentioned that it is quite customary that the computational security parameter is not explicitly stated.
Furthermore, and this is directly related to your initial question, it is important to take into account that a protocol can be computationally secure but also have a statistical security parameter. This can happen, for example, if the protocol makes use of cryptographic tools such as encryption and so on, but in some parts it relies on, for instance, a statistical check that the adversary can cheat on with negligible probability. Since it is still true that the protocol is broken if an adversary has unbounded resources then your conclusion is right: the type of security would be only computational.