The popular misunderstanding is evident in the following joke:
Descartes is sitting in a bar, having a drink. The bartender asks him if he would like another. "I think not," he says, and vanishes.
The current Wikipedia article dismisses this move as sheer logical fallacy (denying the antecedent), but I think there is something deeper going on. We see this when people joke, "I drink, therefore I am", intending their comment to reveal something about their person (e.g. that social drinking is central to their lives, or something they highly value). This indicates that they were interpreting Descartes as claiming that thinking is, in some sense, the reason why he exists. After all, that does sound like something a philosopher might say -- compare Socrates' claim that "the unexamined life is not worth living."
People commonly imbue conditionals and inferences with metaphysical significance. For example, the claim "if it is raining then the grass is wet" naturally suggests a causal relation between the two: it's raining, and that's why the grass is wet. Similarly, one might interpret "I think, therefore I am," as meaning something like, "I think, and that's why I am." This interpretation explains why people are tempted by the converse inference from non-thinking to non-existence. It's not so illogical after all.
But it is a mistake all the same. Descartes isn't claiming here that thought is the reason for his existence. Rather, it's simply an infallible indication of it. An evil demon might deceive you about all manner of ordinary facts, but from the mere fact that you're thinking or having mental experiences at all, it logically follows that you must first exist. You couldn't possibly be mistaken about that, at least, no matter what the demon might try.
Note the crucial distinction between epistemology and metaphysics. Epistemology is about evidence and what we can know. Metaphysics is about reality in itself. The two can easily come apart, as when we imagine a hypothetical scenario involving deception: we imagine that something is true in fact, though nobody knows it because the evidence is misleading. We can similarly imagine someone being psychologically certain that they "know" something, when in fact their belief is not true at all.
Now, the Cogito highlights the logical relation between "I am thinking", and "I exist". If the first is true, this logically guarantees that the second must also be true. This has epistemic significance. But it is not metaphysically loaded. The abstract logical relation says nothing about how the two facts are connected in the structure of reality. In particular, it is not to claim any metaphysical dependence of the second on the first, i.e. that I exist only because I'm thinking, such that if my thinking were to cease then so would my existence.
(N.B. A Cartesian [from Descartes] Dualist will actually agree with the metaphysical thesis that we are essentially mental beings. But this is based on other arguments, not the cogito.)
Disclaimer: I'm no historian, so bear in mind that my purpose here is to clarify the logic of the cogito and related concepts, not to faithfully represent the historical Descartes. Take any claims about the latter with a grain of salt.