Suppose we have a Gaussian Process $X_t$ on $\mathbb{R}^n$ with mean function $m(t)$ and covariance function $K(t,s)$. Then is $X_t$ being sample continuous (i.e. the sample paths of $X_t$ are almost surely continuous everywhere) equivalent to $X_t$ having a modification that is sample continuous (via for example Kolmogorov's continuity theorem)? I've seen sources use them interchangeably but I don't quite understand why?
If there is a difference, does there exist a condition to ensure $X_t$ itself is sample continuous, and not just having a sample continuous modification? If so, can those conditions be purely expressed in terms of the moments of $X_t$ (like with Kolmogorov's continuity theorem)?
If $X_t$ is sample continuous, does there always exist a modification (which has the same mean and covariance as $X_t$) that is not sample continuous?