In physics generating functions are used to produce symplectomorphisms between special symplectic manifolds. More explicitely, given two manifold $M$ and $N$ with respective local coordinates $q_M$ and $q_N$ (I won't write indices and contractions explicitely), we have a function $F(q_M,q_N)$ on $M\times N$ which allows us to define a diffeomorphism between $T^*M$ and $T^*N$ given in respective coordinate systems $(q_M,p_M)$ and $(q_N,p_N)$ by the implicit equation: $$\begin{align}\tag{1}\label{gen} p_M&=\partial_{q_M}F(q_M,q_N)\\ p_N&=-\partial_{q_N}F(q_M,q_N) \end{align}$$ where the first line defines $q_N(q_M,p_M)$ and the second line $p_N(q_M,p_M)$.
Of course in order to have a diffeomorphism we have to require local conditions like $\partial_{q_N}\partial_{q_M}F$ to be nonsingular as well as global conditions, that we assume satisfied. My question is how to understand the construction of the diffeomorphism from a geometric perspective.
The case I described is quite clear and well documented in the mathematics literature: the quadruplets $(q_M,p_M,q_N,p_N)$ satisfying (\ref{gen}) are given by the graph $\Gamma_{dF}$ of $dF$ in $T^*M\times T^*N$, which has to represent a diffeomorphism from $T^*M$ to $T^*N$. The map is symplectic if and only if $\omega_M-\omega_N$ vanishes when restricted to $T\Gamma_{dF}$, which can be proved to be the case from (\ref{gen}). Actually, on $\Gamma_{dF}$ we have $$ \theta_M-\theta_N = d(F\circ\pi) $$ where $\pi : T^*(M\times N)\mapsto M\times N$ is the bundle projection and $\theta_{M,N}$ the tautological one forms. As a remark, the whole construction depends on $dF$ so that we can talk of generating form (that needs to be closed but not necessarily exact).
Now, we can find different ways of generating a sympectomorphism from a given function. Writing formally $F(p_M,q_N)$ (because from a geometric point of view a element of the cotangent bundle $T^*M$ knows which point of $M$ it projects to), the diffeomorphism is now defined by $$\begin{align}\tag{2}\label{gen2} q_M&=-\partial_{p_M}F(p_M,q_N)\\ p_N&=-\partial_{q_N}F(p_M,q_N) \end{align}$$ using the tangent-cotangent bundle duality. I am looking for a geometric picture of this construction.
At first glance, to even make geometrical sense of (\ref{gen2}), as $\partial_{p_M}F(p_M,q_N)$ would be an element dual to covector, that is, a vector, there needs to be some correspondence between points of $M$ and of the fiber of the tangent bundle that would be somehow trivialized. The framework that seems natural to me would be to assume that $M$ is a vector space, so that $TM\simeq M\times M$ and $T^*M\simeq M^*\times M$. Indeed looking at $dF$ we expect an equation of the form $$dF = \partial_{p_M}Fdp_M + \partial_{q_N}Fdq_N = -q_Mdp_M -p_Ndq_N$$ where the contraction $-q_Mdp_M$ only makes sense in a coordinate independent way if $q_M$ can be interpreted as a vector of $T_{q_M}M$.
Looking at references [Foundations of Mechanics, Abraham and Marsden; Introduction to Mehcanics and Symmetry, Mardsen and Ratiu], I find that they suggest to use an arbitrary one-form $\theta$ such that $d\theta = \omega_M-\omega_N$, which to my understanding would express the way the generating function generates the diffeomorphism (like $-q_Mdp_M -p_Ndq_N$). The generating function would also be actually defined on the graph of the diffeomorphism. Unfortunately from this point of view I don't see a general picture which would include the $F(p_M,q_N)$ generating functions.