I've been reading John Stillwell's translation of the famous Analysis Situs and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the introduction), Stillwell claims Poincaré defines it, in paragraph 14, as a manifold with trivial fundamental group (as is the modern sense of the term). However, in the actual text I find no such definition clearly stated. The term 'simply connected' is first used on page 65, seemingly with no definition given. On page 74 we have something suggestive:
Thus we have three manifolds whose group are of finite order, but non-isomorphic, so the manifolds cannot be homeomorphic. Nevertheless, they have the same Betti numbers $$P_1 = P_2 = 1.$$ It seems natural to restrict the meaning of the term simply connected to manifolds whose group $G$ reduces to a single substitution. Then a closed manifold of more than two dimensions can have a group $G$ of finite order without being simply connected.
I can only interpret this as meaning that, to Poincaré, simply connected meant trivial fundamental group plus something else. It seems to me like this something else is all Betti numbers being equal to 1 (note that Poincaré's Betti numbers differ from the modern ones, being shifted up by 1).
Furthermore, on the very last page of the fifth (and last) complement, Poincaré states his famous conjecture:
Is it possible for the fundamental group of $V$ to reduce to the identity without $V$ being simply connected?
(Poincaré goes on to end the paper claiming, ironically enough, "However, this question would carry us too far away.") This would seem like a silly question if simply connected were a synonym of trivial fundamental group (unless I misunderstood what he means by 'reduce to the identity' -- I am interpreting this as being a trivial group).
Even if understood as I suggested above, this is still a bit strange a question, as it is vastly different from what gets called the Poincaré conjecture nowadays -- in fact, it's easy to show that a simply connected (in the modern understanding of the term) closed 3-manifold is a homology sphere (in particular, has the same Betti numbers as the sphere), and so is simply connected even in the more restrictive definition. Indeed, the proof is just Poincaré duality, which obviously Poincaré knew about, so he couldn't have missed it. Nevertheless, on a footnote Stillwell claims this is a correct statement of Poincaré's conjecture.
How to make light of this confusion? What did Poincaré mean by 'simply connected' and how did he interpret his own conjecture?