I am looking for a full account of the relationship between the various versions of Floer theory on a symplectic manifold $M$. If we take the usual Floer equation (Hamiltonian version) \begin{equation*} \frac{\partial u}{\partial s} + J \left( \frac{\partial u}{\partial t} - X_H(u) \right) = 0 \ , \end{equation*} there are two "natural" types of boundary conditions: periodic conditions $u(s,t) = u(s,t+1)$ on the cylinder, or "fixed ends" $u(s,0) \in L_0$ and $u(s,1) \in L_1$ on the strip where the $L_i$ are Lagrangian submanifolds of $M$. One can always find a new complex structure $\tilde{J}$ with map $v : \mathbb{R}^2 \to M$ and transform the Floer equation into the unperturbed Cauchy-Riemann equation \begin{equation*} \frac{\partial v}{\partial s} + \tilde{J} \frac{\partial v}{\partial t} = 0 \ . \end{equation*} The two sets of boundary conditions then become: periodic $v(s,t) = \phi_H(v(s,t+1))$, or "fixed ends" $v(s,0) \in L_0$ and $v(s,1) \in \phi_H(L_1)$ where $\phi_H$ is the time-1 symplectomorphism generated by $X_H$.

I can roughly see how this infers several relationships between both types of Floer theories, but my understanding of all the possible correspondences is far from complete. In particular, fixed-end boundary conditions identify solutions of Hamiltion's equations with intersections of $L_0$ and $\phi_H(L_1)$. Fine, but what can be said about periodic boundary conditions? Is Lagrangian Floer theory of the intersection of the diagonal with the graph of a Hamiltonian diffeomorphism the only version of Lagrangian Floer theory that has a correspondence with periodic boundary conditions? There is also apparently a way to show that, on a compact manifold, intersections of a Lagrangian with are in one-to-one with critical points of Morse function on the Lagrangian and moreover, that Lagrangian Floer theory of the zero section of $T^*M$ is identified with Morse theory on $M$ (or something along those lines).

I have spent quite a bit of time going around in circles with the literature and so if someone could provide me with a nice reference with where these things are explained in detail (or even perhaps willing to describe some themselves) it would be very useful.

Thanks.