Descartes (and a tangent on some definitions of 'person')

Descartes is a rather controversial philosopher, at least among conservative Catholic circles. Presently, I have read his Discourses, I-III and the first half of IV, and his Meditations, I, II, & III. I would like to explain his "I think, therefore I am".

"I Think, Therefore I Am"
That is THE statement of Descartes, and few know much more about him than that. At first this statement seems to say that thought is the source of existence. But let us examine the text, keeping always in mind that Descartes was rather obsessed with doubting everything until he had undeniable proof for it.

"[F]rom the very fact that I thought of doubting the truth of other things, it followed very evidently and very certainly that I existed[...][T]here is nothing at all in this 'I think therefore I am' that assures me that I am speaking the truth, except that I see very clearly that, in order to think, it is necessary to exist."

You see, he is so concerned about not calling true anything that has the slightest possibility of being false, that the only thing left for him to know for sure is that his mind, which he calls 'I', exists. His proof for the existence of his mind is that it is thinking. And so, his 'I think, therefore I am' is really just a poor way of saying 'Because I think, I know I (that is, my mind) exists.

This would all be well and good, except for what he says during the ellipsis in the above quote.
He states:

"had I simply stopped thinking [...] I would have no reason to believe that I had existed."

This statement gives two criteria for a thing, specifically a human, to be considered living:
1. The thing must be thinking
2.The thing must be conscious of its thinking.

True as it may be that one's thoughts are a proof to oneself of one's existence, they are not a criteria for being considered a living human.

Speaking of criteria to be considered a living human, these are the definitions of some other philosophers of the word 'Person'.

Soul trapped in a Body - Plato
Rational Animal - Aristotle
Embodied Spirit - St. Thomas Aquinas
Thinking Thing - Descartes

It seems that these Philosophers fall into 2 groups- Those that define a person by what it does (thinks), and those that define a person by what it is made of (a body and a soul).