But the language is such that thinking is not a constant, i.e., not an every-present state (even when awake). As @Andy claims that Bhudda proposes, “the self has no inherent existence.” (And here I wonder if/how our Bhuddist dissatisfaction (dukkha) is similar to Descartes’ doubt, hiding his desire for certainty).
And we do not “decide” to think (you may still be picturing thought as our mental self-talk), but an expression is judged to be thought on our shared criteria.
Now I grant that consciousness is being awake (or aware), as opposed to sleep (or oblivious). And Descartes does point out that even if I am being deceived, I am “not nothing”, but that state does not meet his criteria (certainty, without doubt, as Rovelli starts with, by @Andy). Now I can be said to be without doubt—but not based on being convinced (persuaded)—but in having conviction (self-assurance). I am resolute, which is not to say this is a ground or justification (just bull-headedness), but it is a readiness to be intelligible for myself. So it is not proof ahead of time, but acceptance of ongoing responsibility.
Well, first, It is described as a “proposition”, which is a claim, but not a “statement” (as I would state a fact, which would be verified). It is not up for consideration; it is proposed in the sense of “put forward by me”, as one would create a law, i.e., the sense of “affirming”, as you put it, as in a declaration.
Not to side-step this, but I would suggest that simplifying to “exist or not” cuts off the possibility of learning how it works in each case and the more nuanced similarities and distinctions. I would say that imagining a third place or state is to lock ourselves into frameworks we do not then give ourselves a chance to understand, or learn from.
Back finally to the matter at hand, and without, again, any real knowledge of Buddhism, I would suggest that Descartes takes the self as something brought about in thought, or intelligibility, as an active process. Maybe the Bhudda is suggesting that the self is “formed”, as you say, passively, in acceptance, or in the “sati” moment (as @Wayfarer might expound on) of letting go of what is now called the ego (the sense of an inherent self, as @Andy puts it). Off the cuff, I would again say that Descartes origination of the self is not the ego either (our sense of an ever-present, casual, or through-line self).
But are they so disimilar? In that Descartes’ self is resolute in its acceptance of the responsibility for the intelligibility for my self, and the Bhudda in a sense could also be said to be resolute in not accepting what does not originate from me, ignoring the desires and limits given by the world.
