NB: Continental philosophy is a little outside my wheelhouse, but I'll attempt to respond to the question as I believe you have asked it.
You ask:
So, to close, is it the case that "thought" is distinct from "thinking", and if so, can we finally admit that we may deny we are entirely responsible for the latter?
(The TLDR is yes.)
Georg Wilhelm Friedrich Hegel is one of the big H's to follow Kant and to continue the work of phenomenology moving in a different direction from Classical Empiricism and Classical Rationalism. The latter point is important because phenomenology attempts to reconcile, much in the spirit of transcendental idealism, a number of philosophical themes such as the senses, introspection, and postivism under a banner.
Descartes did not believe animals were capable of thought, and Cartesian duality meant that without thinking, there could be no mind; Descartes believed on this account animals had no soul. See this answer here and read the comments below for related back and forth on the relationship between consciousness and thinking (PhilSE). Of interest to your questions are the notions of self-reflection, self-awareness and self-consciousness. From the other PhilSE answer's comment:
According to "Consciousness" (EoP), "Descartes... always described the states we now call conscious as states that one is conscious of... because [believing introspection was complete], they saw no need to use the term conscious to mark a distinction." The article goes on to raise the conscious/unconscious dichotomy recognized later under Freud, etc. Which is consistent with Cartesianism and rationalism more broadly construed. - JD
My personal belief is that Kant is the most famous of the more modern era, contra the Ancient Greeks and Scholastics, where philosophy mind really begins to open the door to the intricacies of cognition, and Hegel was part of the movement and was teasing out the vocabulary to describe more modern conceptions. This post-Kantian German idealism was fond of grand, sweeping metaphysical speculation. According to "Hegel: Social and Political Thought" (IEP):
The Phenomenology of Spirit (Die Ph?nomenologie des Geistes), published in 1807... provides what can be called a “biography of spirit,” i.e., an account of the development of consciousness and self-consciousness in the context of some central epistemological, anthropological and cultural themes of human history. It has continuity with the works discussed above in examining the development of the human mind in relation to human experience but is more wide-ranging in also addressing fundamental questions about the meaning of perceiving, knowing, and other cognitive activities as well as of the nature of reason and reality. [emphasis mine]
So, we have set the stage to answer the question by showing that under rationalism, the nature of psychology (which didn't really take off until the 19th century) was more primitive than post-Kantian thinking. Whereas Descartes had a crude map of the mind and placed a great faith in introspection in his Meditations, Kant had really moved the model of cognition forward with his various Critiques and moral philosophy. Hegel, who sought to be original would have anticipated later developments by Husserl and Heidegger among others.
We can jump over to the SEP for some analysis of Hegel's work. Hegel, who was proud of his speculative system of which the Phenomenology of Spirit was intended to be a introduction obviously is a complex topic. From "Georg Wilhelm Friedrich Hegel" (SEP):
In his 2019 monograph devoted to Hegel’s Science of Logic, Pippin stresses that while Hegel has a metaphysics, it is a metaphysics "in a new sense" (Pippin 2019, 5). While Hegel’s logic is a "science of pure thinking", there is, nevertheless, an "identity (a ‘speculative identity’ to be sure) between the forms of pure thinking and the forms of being" (2019, 6), that is, between logic and ontology. Hegel’s fundamental link to Kant had been his (Sellarsian) idea that "thinking is discursive; to think what is the case is to assert that it is (the basic unit of intelligibility in Hegel’s account is the judgment; assertion its linguistic manifestation)" (2019, 14). However, Hegel was critical of those residual empiricist elements within Kant in which sensory intuition played the role of providing a definitive and unrevisable content for logically structured judgment forms—elements criticised by Sellars with the notion of the "myth of the given" (Sellars 1977) [emphasis mine]
Thus, according this analysis, Hegel's view on thinking was anticipating post-postivist skepticism of the infallible nature of perception and observation that essentially killed logical positivism and empiricism. If this analysis of Hegel is correct, it would have to be because Hegel understood that our representations are fallible. And if those representations are fallible, it would mean that there is thinking, and then there is "having thoughts", something akin to self-awareness understood as self-critical analysis of one's thoughts, in contradistinction to simple awareness. This would anticipate more modern phenomenological debate like the notion of phenomenological reduction (IEP).
So, if we take Pippin's analysis above, we see that Hegel believe that there is discursive thinking, which entails both judgement and ontological commitment in the modern sense, and a skepticism of the truth of what Kant called the spontaneity of consciousness. Thus, there is thinking, but there is also incumbent upon the thinker an analysis of sort, presumably the sort of philosophical analysis Hegel himself was engaged in.
If Pippin's analysis is that Hegel anticipated Sellar's and his Myth of the Given, then absolutely, the often spontaneous content of thought would be held very distinct from the process of thinking. This would be in line with the very ancient notion of the difference between appearance and actuality. Of course, the terminology is still very contentious. Some thinkers like Searle reject the representational theory of mind instead offering his direct realism. The debate among phenomenologists over terminology is substantial enough I've avoided it as an analytical thinker. But Hegel's fallibilism would entail that mere thinking is insufficient to be a science of spirit, and instead, one would have to look more deeply inwards towards personal narratives and more intensely outwards for meta-narratives like Fichte.
