It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts.
Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy functional as the Morse function, then critical points are geodesics and the index is the index is computed via the second variation formula. As explained in Milnor's classic book, one can recover the homology of the loop space by this technique.
Case 2: On the other hand, when one does Morse theory on the space of (unbased) contractible loops of a symplectic manifold endowed with a Hamiltonian H by using the symplectic action functional, then Floer famously showed that (under suitable hypotheses) one recovers the singular homology of the underlying symplectic manifold.
It seems to me that Case 1 can be interpreted as an infinite dimensional generalization of finite dimensional Morse theory in a very straightforward way. However, Case 2 seems to be of a different nature since the Morse function is not computing the homology of the space on which this function is defined.
Is there an intuitive way to explain this difference? Given a functional on an infinite dimensional space, can we reasonably "guess" what kind of homology theory we will get by "doing Morse theory" with this functional?