classical mechanics - What is the function type of the generalized momentum?

Let

$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$

denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action

$$S[q]=\int_{t_1}^{t_2} L(q(t),q'(t),t)\, \mathrm{d}t,$$

the first $n$ slots are evaluates at a path $q:{\mathbb R}\to {\mathbb R}^n$, the second $n$ at $q'$ and the last one is for possible explicit time dependencies.

Are generalized momenta defined as functions of the generalized coordinates, i.e.

$$p_j=\frac{\partial L(x^1,\dots,x^n,v^1,\dots,v^n,t)}{\partial v^j},$$

or as associated with a curve $q:{\mathbb R}\to {\mathbb R}^n$, i.e.

$$p_j=\left.\frac{\partial L(q^1,\dots,q^n,v^1,\dots,v^n,t)}{\partial v^j}\right\rvert_{\large{q=q(t),\ v=q'(t)}},$$ ?

In the latter case it's a function of time only, and $q$ is buried somewhere within it.

A question that is codependent with this might be: What is the type of the total derivative and the term $\frac{\partial \mathcal{L}}{\partial q_i}$ in differential equation in $q:{\mathbb R}\to {\mathbb R}^n$, which is usually expressed as

$$\frac{\mathrm{d}}{\mathrm{d}t} \dfrac{\partial \mathcal{L}}{\partial {\dot q_i}} - \frac{\partial \mathcal{L}}{\partial q_i} = 0.$$