Clarification of the sample space of stochastic processes

I have encounter two different definitions of a stochastic process. The first definition tells me every outcome is associated with an entire deterministic function of t (every outcome maps to one sample path in the ensemble). A stochastic process can therefore be regarded as a S^T-valued random variable, and the sample space is the sample space of that random variable. Fine.

The second definition tells me to consider a stochastic process as a sequence of random variables on a probability space (Ω,F,P). One outcome in Ω results in a mapping for each and every random variable in the sequence (such that a "deterministic" function of time is created). Fine.

What I would like to have clarified is if the two definitions always refer to the exact same "sample space"? The first definition refers to the sample space of the S^T-valued random variable, and, the second definition refers to the sample space shared by all the random variables in the sequence. Are these sample spaces the same (always)? Is it called the sample space of the stochastic process?

If $$(X_n)$$ is a sequence of random variables on $$(\Omega,\mathcal F,P)$$ then we can define measurable function $$X$$ on the same probabilty space with values in $$\mathbb R^{\mathbb N}$$ by $$X(\omega)=(X_n(\omega))$$[ This is measurable when $$\mathbb R^{\mathbb N}$$ is given the Borel sigma field). Conversely. any measurable function $$X:(\Omega,\mathcal F,P) \to \mathbb R^{\mathbb N}$$ defines a sequence of random variables $$(X_n)$$ by defining $$X_n(\omega)$$ as the n-th coordinate of $$X(\omega)$$. There is no need to change the sample space $$(\Omega,\mathcal F,P)$$ for this association.