In this math.overflow question: https://mathoverflow.net/questions/143116/where-does-the-notation-pi-1x-x-for-the-fundamental-group-first-appear, an answer posits that Poincaré regarded the fundamental group "as a group of permutations of fundamental regions". In this History of Math & Science question: https://hsm.stackexchange.com/questions/2369/how-did-poincar%C3%A9-discover-the-fundamental-group, an answer posts Poincaré's writing from his Analysis Situs on the definition of the fundamental group.
However, I do not understand what Poincaré is saying (or what "group of permutations of fundamental regions" means); so I wonder if someone can explain what Poincaré is trying to say in this passage.
Another issue I have is about the chronology. In the Wikipedia article for the Poincaré conjecture: https://en.wikipedia.org/w/index.php?title=Poincar%C3%A9_conjecture&oldid=1044860877#Poincar%C3%A9's_question, it makes it seem like Poincaré developed the theory of the fundamental group after his discovery of the Poincaré homology sphere in 1904. However, Analysis Situs was published/written in 1895, in which the above excerpt on the fundamental group is taken.
Did Poincaré develop the theory of the fundamental group in response to the homology sphere, and if not, was there a specific problem that motivated his discovery of the fundamental group?
I am aware of a similar MSE question: What was Poincaré's original idea of the fundamental group?, but the answers were "link-only" and didn't answer my questions.
Finally, even if people do not have specific answers to the above questions,
I would appreciate if someone knows what could have motivated Poincaré to consider the idea of the fundamental group, i.e. a priori why would it be a good avenue of investigation to study loops/paths in a space to better understand the topology of that space.**
EDIT: I will try to better explain my last question. I know for example that in math studying maps out of or in to a space illuminates some properties of that space; but for the fundamental group we are considering maps from $[0,1]$ in to the space (or for homotopies maps from $[0,1]\times [0,1]$ in to the space). This to me seems somewhat arbitrary: why should the unit interval and square be the “basis” on which we study other spaces?