You seem to have some funny interpretations of the words "think" and "believe". This is hindering our ability to communicate with one another. For instance, I interpreted your first question as a petty insult. Thus, my answer might not make any sense, as I'm not entirely certain of the definitions you are using.
I was claiming that in Descartes "I think therefore I am" is not a fundamental belief, as you claimed, and tried to explain why. There is of course more: when I say "You don't think, do you?" strictly Cartesian answer would be that this is impossible, because he maintains that every human is endowed with thinking facilities. Belief is then in his scheme of things of secondary importance compared to thinking, as far as we are concerned with true knowledge of things. Which is different from your thoughts on the subject:
Ok. Now, my question to you is simply: do you believe in this or do you think that?
I would say that both thought and belief are fundamental in their own ways. If you have no thought, then you cannot reason how the world might behave, and therefore have nothing to believe in. If you have no belief... none whatsoever... then you cannot believe what your own senses tell you, and you have no foundation upon which to reason. This principle expands further to not being able to believe anything which might confirm or deny the results of your reasoning. Thus, a being without belief is unable to reason, and a being without reason lacks belief. That is what I think.
where, if I understand it at all, both belief and thought cannot do one without other. Now back to Descartes:
... one can believe in anything at all
There are a great many things out there that people believe in. Some exist in an absolute sense, while others do not. This is the quest of science, to ascertain the difference. There is no clear indication if this is a finite task.
In the Cartesian project this task would be endless.
(This is also why fiction such as The Matrix, or certain Star Trek episodes featuring the holodeck have such appeal. They ask us to re-examine our most basic beliefs, if only for a moment. If one can suspend disbelief in those contexts, anything is possible. In the context of fiction, that is quite desirable.)
In Matrix there is that pill-choice given, and the Cartesian way would be to choose going down the rabbit hole and figure out how the system works.
I hold, though, that all logic must be based upon fundamental axioms. If the axioms we choose are true, and our logic is sound, then we may deduce further truth. If our axioms are false, then we have no assurance that we will gain truth through reason. Any attempt to build a framework of reason that is not built upon axioms will either be tautological (hidden axioms), or self-contradictory. Thus correct belief with thought leads to further correct belief. No amount of thought will lead to truth if our core beliefs are wrong. That is what I believe.
Cartesian position is here much more radical: mathematics works regardless whether I believe [in] it or not, thus it is the way toward certainty.