Not to downplay Descartes (his philosophy, specifically the process he undertakes, is worth a study), but something that the pop summaries of his writing really gloss over is that his entire premise hinges on something he takes as assumed true that need not be, and if it isn't, the rest goes straight off the rails.

Paraphrasing from memory: he pins "cogito ergo sum" on the assumption that objective knowledge is possible. That assumption is built atop imagining an alternative: an evil demon has full control over his senses and feeds him whatever it wants to (essentially, the 'Matrix hypothesis'). His solution to this concern? "A loving God wouldn't create a universe that banal and meaningless."

It's only a proof if you accept the postulate that objective reality is correlated with experience.

But how do you even distinguish appearance of thought from actual thought, objectively? Of course you can always argue that actual thought is something that is subjectively experienced, but that leads to solipsism, doesn't it?

Solipsism is what you get from accepting the Cogito "I know I exist for sure, and only my own existence can be verified by the cogito"

To answer your question, I claim that you can't. Radical skepticism ("we live in the universe of the evil demon"), which is what I advocate for, means that you can't even claim that you exist. See the works of Max Stirner for further elaboration of the implications of this for Philosophy.

This is essentially the philosophical zombie thought experiment: https://en.wikipedia.org/wiki/Philosophical_zombie