Want is a transitive verb in English and does not make sense if it is used without a direct object.
Both of your examples have direct objects.
I want to be famous.
Direct object: "to be famous"
"To be famous" is an infinitive phrase and infinitive phrases are a sub-category of noun phrases. Therefore, the entire phrase can be thought of as a noun. As far as English is concerned, in the following sentences,
I want to be famous
I want a cat
"cat" and "to be famous" serve the same function.
In your second example, "to be famous" is still the direct object. "You" is something called the indirect object.
It seems that nobody is going to answer, so let me try.
It's a bit difficult too for me to understand the difference but let me simplify your example.
I think that we can represent the same meaning with two different structures.
(1) The future worries me
(2) I worry about the future
According to Cambridge Dictionary
In the first case, we got
worry [TRANSITIVE]
to make someone feel unhappy and frightened because of problems or
unpleasant things that might happen
In the second case, we got
worry [INTRANSITIVE]
to think about problems or unpleasant things that might happen in a
way that makes you feel unhappy and frightened
Let's take a look at Oxford Dictionary
Feel or cause to feel anxious or troubled about actual or potential
problems.
[no object] he worried about his soldier sons in the war
[with object] there was no need to worry her
Feel [INTRANSITIVE]
Cause to feel [TRANSITIVE]
Knowing all this, let's return to your example
People have a variety of factors in their lives that cause them to
worry
[Simplifying] A variety of factors worry them. [TRANSITIVE]
I found myself worrying about the tests
[Simplifiying] I worry about the test [INTRANSITIVE]
about the test is NOT the object, it is a prepositional-clause.
Best Answer
No, it's not transitive in that context. "chair" is part of a prepositional clause that indicates where he sits.
You can check this link from Oxford Learners Dictionary with multiple examples of both the transitive and the intransitive modes but let me quote the definitions:
[intransitive]
[transitive]
Notice that sit is transitive when the subject executes the action over somebody else, a different person.
He sit her child on the chair.
He - subject
her child - object
In your example he sits "himself", that "himself" is a sort of omitted and redundant object, and as you can read in the above definition, that mode is considered intransitive.