Exactly! You are not what you think (or at least, you can’t prove what you are with any certainty). The point of the argument is that the thoughts you have must have some source, whatever it might be.
You are correct that the ‘me’ that my thoughts are coming from cannot be found or percieved directly. (I do seem to be able to think of it, albeit speculatively.) It does have an observable by-product though: the thoughts themselves. Where there’s smoke, there’s fire, so to speak.
The use of the word “I” is just barely more than a convenience: since the only thoughts I can be aware of are the ones that I myself am having, I arbitrarily define them as ‘mine’, a product of ‘I’. And so, even though I have absolutely no knowledge about the actual properties of the thinker, I choose to label it ‘I’ because I have already claimed its thoughts as mine. If you feel this labeling to be a bit presumptious, okay, but the thought experiment has still yielded some result, which is pretty much what dalovindj said: “I think/feel, therefore something exists.”
I think perhaps that poor little ol’ cogito ergo sum has been burdened with more assumed meaning than it actually has the power to back up. It is the commentary on Descartes’s failure to justify the denial of the existence of his own thoughts.
[I wish I had been this clever then]
Prof: “So are you your body or your thoughts?”
Me: “No. Neither.”
Prof: “How so?”
Me: “I wear my body and produce my thoughts. Neither are me.”
Prof: “What are you then.”
Me: “I haven’t the faintest idea, except that it’s able to produce my thoughts.”
[/I wish I had been this clever then]