The question is actually very simple:
Could I define the stochastic process $X=(X_t)_{t\in J}$ from $(\Omega,\mathcal F, \mathbb P)$ into $(E, \mathcal E)$, as a sequence of random variable?
What I mean is, there is some difference between a stochastic process and a sequence of random variable?
I see that the question is quite silly, and in my opinion they are the same thing, but I would like to be sure. Thank you for the patience.
Actually this viewpoint is very adequate because we often see e.g. Random Walks with $J = \mathbb{N}$ under this exact idea with the sequence interpreted as a timeline. Notice however that you are limiting yourself to certain subsets $J$ as we are losing important aspects of a sequence otherwise. If you ment sequence rather in a more describing way we still get (and use) this perspective for example with the Brownian Motion which is a stochastic process with $J = \mathbb{R}_+$ as a stochastic process in continuous time.