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.

I think the idea is because Gaussian Processes are only uniquely defined up to the mean and covariance and therefore not unique (is this correct?), you can 'choose' to define your process to be the version/modification that is sample continuous, but is there a more formal way of saying this?

In general (i.e. not necessarily just for Gaussian Processes) when is having a sample continuous modification equivalent to sample continuity? Clearly if $X_t$ was sample continuous, you can easily find a modification $Y_t$ that is sample discontinuous (see for example https://math.stackexchange.com/questions/2608352/discontinuous-brownian-motion).