So if there was an act of thinking, isn't it necessary to assume that there was something engaged in this act, or else it wouldn't be an act at all? Why not call this thing "I"? Of course it is impossible to doubt the existence of somebody thinking, and we can even call it “I”. The question is the degree of dependence of this process. How is it determined and conditioned?? “Existence really is an imperfect tense that never becomes a present”

Nevertheless,psychoanalisys practices demonstrate grounding procedures for establishing "I"(In the absence of Descartes' style methods)
I would bring you Felix Guattari with
his theory of machinic unconsciousness. But even without referring to Guattari, in my personal life I've never met anybody thinking independently in cogito's manner. So, who or what is the source and
the reason of "thinking I"? We are no more as terminals in complicated machinic assemblages!

That said, it is not just any synthetic a priori truth; it is rather the I's self-grounding act, its self-creation. Descartes explains the nature of such self-grounding judgments in the Replies: "We cannot doubt them unless we think of them, but we cannot think of them without at the same time believing that they are true..., that is, we can never doubt them." (The Theological Origins of Modernity, Gillespie, (196-197)).
I think that is why many of us again and again coming back to Descartes: we try to find some new ground
where we lost all possible grounds, in our thinking or in its absence. That is what Nietzsche did, even when he tried to dismantle Descartes's cogito.

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"It is impossible that a thought could will itself into existence. It must already exist in order to will anything. So willing itself into existence would mean that it wills itself before it exists, to bring itself into existence. But this is impossible because it would mean that it exists prior to its own existence. This description of a thought is nothing other than a description of a self-caused thing. That's contradiction because it means that the thing must both exist (as the cause) and not exist (prior to the thing's existence) at the same time."
It would be a paradox if Descartes was completely isolated. However, he supported himself by prevailing contemporary discourse ( with all possible connotations) of his time.

However the association of the "I" with this thought, is not equally affirmed by Descartes, indeed throughout the Meditations Descartes refers to himself as "what am I only a thing that thinks". There is a significant distance between the concepts of :

1) a thing that thinks
2) a thing that experiences thought
3) a thingless experience of thought
4) an 'I' thinking

From my own reading of Descartes I fail to see how anything more than the assertion at 3, a thingless experience of thought has been effectively reasoned by Descartes."

Could you illustrate all 4 concepts by concrete examples? Who developed them? And if not Descartes,
who was first to introduce the concept of the thinking "I'?

We can consider the most prominent figures of psychoanalyses: Freud and Lacan.
Their theories and practices can provide evidence that Nietzsche was right: the thought is determined by anonymous and non-personal factors, such as Unconsciousness of Freud and Real of Lacan

Neitzsche in aphorism 17 (BG&E) Reiterates Descartes own criticism of himself, with the empirically correct observation that 'a thought comes when it wills' and not when this 'I' thing wills it. Therefore if thought simply comes when it wills, and is not generated by the entirely presumptive 'I', we must conclude that thought is independent of the 'I' and return to the fundamental principle that thought exists apriori." Nietzsche was absolutely right, but it is still vague what does it mean that thought comes when it wills and exists priori. It is still opened to so many interpretations. Definitely, Deleuse furthermore develops this idea of Nietzsche when he splits the thinking subject into two mutually dependently acting subjects: the subject of the enunciation and the subject of the statement, the thinking "I", and existing "I". Both kinds of "I' are related through the dominating (in particular philosophical) discourse.

"The cogito comes in different forms. "I think, therefore I am." "This proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind."
Descartes established the fundumental doubling of the subject into two different, but coexisting
subjects: the subject of the enunciation and the subject of the statement.