These are all standard TQFT structures which reflect the combinatorics of Riemann surfaces. In general, one starts with a compact Riemann surface (possibly with boundary $\bar{S}$ and a finite set of marked points $\Sigma\subset\bar{S}$. There are four kinds of marked points in the set $\Sigma$. First, there is an essential difference between a boundary marked point $\zeta\in\Sigma\cap\partial\bar{S}$ and an interior marked point $\zeta'\in\bar{S}\setminus\Sigma$. Second, $\Sigma=\Sigma^{in}\cup\Sigma^{out}$ is partitioned into inputs and outputs. Formally, a connected component $C\subset\partial\bar{S}\cap\Sigma$ is mapped to a Lagrangian label $L_C$ and a boundary marked point $\zeta\in\bar{S}\cap\Sigma$ is required to converge to an intersection point between two nearby Lagrangian submanifolds $L_{\zeta,0}$ and $L_{\zeta,1}$. On the other hand, every interior marked point is a mark served to break up the symmetry when treating moduli problems or should be mapped to a period orbit of some Hamiltonian vector field $X_H$.

Let me illustrate the above formalism with some elementary examples. When $\bar{S}=\mathbb{D}$ is the closed unit disc, and $\Sigma\subset\partial\mathbb{D}$ consists of two distinct points, we get the Lagrangian Floer cohomology $HF^\ast(L_0,L_1)$, where $L_0$ and $L_1$ are the two Lagrangian labels correspond to two connected components of $\partial\mathbb{D}\setminus\Sigma$. When $\bar{S}=\mathbb{CP}^1$ and $\Sigma\subset\mathbb{CP}^1$ consists of two distinct points, one gets essentially the Hamiltonian Floer cohomology $HF^\ast(\lambda H)$, where $H$ is a Hamiltonian function on some symplectic manifold $M$. Strictly speaking, in the latter case we need to decorate $\bar{S}$ with a closed 1-form $\gamma$ so that $\int_{C_1}\gamma=\int_{C_2}\gamma=\lambda$, where $C_1,C_2\subset\mathbb{CP}^1\setminus\Sigma$ are two small circles centered at the interior marked points. Also, we need to fix choices of directions near the two punctures $\zeta,\zeta'\in\mathbb{CP}^1$ to indicate the difference between an input and an output.

The above formalism can be generalized to families of Riemann surfaces define various operations on Floer cochains, which lead to the so-called Fukaya $A_\infty$ structures, and more generally, the open-closed Seidel maps. The definition of the former one involves only boundary marked points on a disc, but the construction of the latter one involves marked points of both interior and boundary types. The associated Floer equations are similar as the ones defining usual Hamiltonian or Lagrangian Floer homologies.

These Floer theories all in principle all Morse theory associated to some action functional defined on an infinite-dimensonal space, that's why certain assumptions or hard works are needed to make them rigorous. In some cases, these theories can be reduced to usual Morse theories, see for example, the work of Fukaya-Oh (https://www.math.kyoto-u.ac.jp/~fukaya/FO.pdf) in the case of Lagrangian Floer theory. But this is not exactly true in general. For example, when $L_0=L_1=L$ are weakly unobstructed Lagrangian submanifolds of $M$, the work of Biran-Cornea (http://www.dms.umontreal.ca/~cornea/qrel.pdf) shows that one has to deform the usual Morse differential using perals (disc bubblings connected by Morse trajectories) to make things correct, which makes it similar to the case of quantum cohomologies.

These are all standard TQFT structures which reflect the combinatorics of Riemann surfaces. In general, one starts with a compact Riemann surface (possibly with boundary $\bar{S}$ and a finite set of marked points $\Sigma\subset\bar{S}$. There are four kinds of marked points in the set $\Sigma$. First, there is an essential difference between a boundary marked point $\zeta\in\Sigma\cap\partial\bar{S}$ and an interior marked point $\zeta'\in\bar{S}\setminus\Sigma$. Second, $\Sigma=\Sigma^{in}\cup\Sigma^{out}$ is partitioned into inputs and outputs. Formally, a connected component $C\subset\partial\bar{S}\cap\Sigma$ is mapped to a Lagrangian label $L_C$ and a boundary marked point $\zeta\in\bar{S}\cap\Sigma$ is required to converge to an intersection point between two nearby Lagrangian submanifolds $L_{\zeta,0}$ and $L_{\zeta,1}$. On the other hand, every interior marked point is a mark served to break up the symmetry when treating moduli problems or should be mapped to a period orbit of some Hamiltonian vector field $X_H$.

Let me illustrate the above formalism with some elementary examples. When $\bar{S}=\mathbb{D}$ is the closed unit disc, and $\Sigma\subset\partial\mathbb{D}$ consists of two distinct points, we get the Lagrangian Floer cohomology $HF^\ast(L_0,L_1)$, where $L_0$ and $L_1$ are the two Lagrangian labels correspond to two connected components of $\partial\mathbb{D}\setminus\Sigma$. When $\bar{S}=\mathbb{CP}^1$ and $\Sigma\subset\mathbb{CP}^1$ consists of two distinct points, one gets essentially the Hamiltonian Floer cohomology $HF^\ast(\lambda H)$, where $H$ is a Hamiltonian function on some symplectic manifold $M$. Strictly speaking, in the latter case we need to decorate $\bar{S}$ with a closed 1-form $\gamma$ so that $\int_{C_1}\gamma=\int_{C_2}\gamma=\lambda$, where $C_1,C_2\subset\mathbb{CP}^1\setminus\Sigma$ are two small circles centered at the interior marked points. Also, we need to fix choices of directions near the two punctures $\zeta,\zeta'\in\mathbb{CP}^1$ to indicate the difference between an input and an output.

The above formalism can be generalized to families of Riemann surfaces define various operations on Floer cochains, which lead to the so-called Fukaya $A_\infty$ structures, and more generally, the open-closed Seidel maps. The definition of the former one involves only boundary marked points on a disc, but the construction of the latter one involves marked points of both interior and boundary types. The associated Floer equations are similar as the ones defining usual Hamiltonian or Lagrangian Floer homologies.

These Floer theories all in principle all Morse theory associated to some action functional defined on an infinite-dimensonal space, that's why certain assumptions or hard works are needed to make them rigorous. In some cases, these theories can be reduced to usual Morse theories, see for example, the work of Fukaya-Oh (https://www.math.kyoto-u.ac.jp/~fukaya/FO.pdf) in the case of Lagrangian Floer theory. But this is not exactly true in general. For example, when $L_0=L_1=L$ are weakly unobstructed Lagrangian submanifolds of $M$, the work of Biran-Cornea (http://www.dms.umontreal.ca/~cornea/qrel.pdf) shows that one has to deform the usual Morse differential using pearls (disc bubblings connected by Morse trajectories) to make things correct, which makes it similar to the case of quantum cohomologies.