Here are your two hypotheses:
A. When followed by 'Object (O) + to be + Object Complement (OC)', cognitive verbs such as think, believe, consider, suppose, understand, imagine, etc. should use a noun phrase (NP) or an adjective phrase (AdjP) as OC.
B. In the structure 'cognitive verb + O + to be + OC', to be cannot be omitted when OC = NP, but can be omitted when OC = AdjP.
Before answering the question, let me make it clear that hypotheses A and B are not terribly accurate. Firstly, those verbs are so called not because of their syntactic behaviors but because of their semantics. Just because you can lump them together under the same semantic group doesn't mean that their syntactic behavior will be the same.
For example, think can rarely be followed by 'O + to be + OC':
(1) ?She thought him to be a teacher.
(1) is unidiomatic at best and can be thought of as a counterexample of hypothesis A.
Hypothesis B presumes that the structure with to be is somehow more readily allowed than the one without. But that presumption is wrong in the case of think:
(1') She thought him a teacher.
Although (1) is ill-formed, (1') is well-formed albeit less idiomatic than (1''):
(1'') She thought he was a teacher.
Likewise:
(2) ?She thought him to be rich.
(2') She thought him rich.
(2'') She thought he was rich.
So it's not like that the structure with to be is always possible and that you only have to figure out when you can omit to be.
Having said all that, in order to answer your question, let's just focus on those verbs that allow the structure 'cognitive verb + O + to be + OC' with or without to be.
In (3)-(5) and (3')-(5'), an AdjP can be OC, but an NP can be OC unless the NP is referential (i.e., referring to a particular person, thing, etc.)
(3) She considers him to be rich/a teacher/Steve Jobs.
(4) She supposes him to be rich/a teacher/Steve Jobs.
(5) She imagines him to be rich/a teacher/Steve Jobs.
(3') She considers him rich/a teacher/??Steve Jobs.
(4') She supposes him rich/a teacher/??Steve Jobs.
(5') She imagines him rich/a teacher/??Steve Jobs.
Since Steve Jobs refers to a particular person, to be is required.