He doesn't implies that. You really don't understand much at all. I already told you. It is a definition. Not an inference. He just called his thoughts "I".
You posted this, bolding mine: "The Cogito, then, is the expression of his conclusion that he cannot similarly doubt the existence of his own mind, because doubting is indeed thinking and
thinking implies the existence of the thinking thing, the "I" of the Cogito"
And in that he declares the existence of a thinking thing. He even declares it to be a self of some kind - identifies it as "I". That's how all read it - including you, as quoted.
- - -
How would you go about proving it is a "mistake in reasoning" since he called thinking "a thinking thing"?
By noting that he didn't observe a "thinking thing", but thoughts. Or by noting that thoughts don't think. Or by noting that thinking is not a thinking thing - thinking is what a thinking thing, if there is one, does.
Running doesn't run. Eating doesn't eat. Remembering doesn't remember. Calculating doesn't calculate.
By the "I", he meant the thinking thing, the thoughts themselves.
Which he assumed without justification, apparently taking it as self evident.
Thoughts do not necessarily make up a "thing" - and definitely are not a thinking thing.
A thinking thing is not its thoughts.
Thoughts do not think themselves.
And if he didn't mean "I" as a term for a self of some kind, he sure picked a strange label for whatever he did mean. Any other letter would have been clearer.
If there is thinking, then thinking is a thing and it's a thing that exists.
Not necessarily. One can assume that, of course, as he did, but it doesn't necessarily follow.
Meanwhile, if such a "thing" is assumed, it doesn't itself think. Thoughts don't think - individually or collectively.
Also on point: if there isn't "thinking", but only thoughts, the entire approach vanishes immediately.