Understanding the sample space and probability distribution of a Gaussian process

I'm trying to unpack the mathematical definition of a Gaussian process, by applying it to an example problem of modelling the height $$h(t)$$ of the tide at time $$t$$.

Let's begin with two definitions of a Gaussian process:

A Gaussian process is a stochastic process $$X_t$$ such that for every finite set of indices $$t_1,\ldots,t_k$$ in the index set $$T$$, $$(X_{t_1},\ldots,X_{t_k})$$ is a multivariate Gaussian random variable.

From Gaussian Processes for Machine Learning by Rasmussen & Williams:

Definition 2.1 A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution.

A Gaussian process is completely specified by its mean function and covariance function: $$m(\mathbf{x})=\mathbb{E}[f(\mathbf{x})],$$ $$k(\mathbf{x},\mathbf{x}')=\mathbb{E}[(f(\mathbf{x})-m(\mathbf{x}))(f(\mathbf{x}')-m(\mathbf{x}'))].$$

From the definition of a stochastic process, for each $$t$$, $$X_t$$ is a measurable function from a probability space to the reals: $$X_t:(\Omega,\mathcal{F},P)\rightarrow R$$ It seems to me that taking $$k=1$$ in the multivariate Gaussian condition above, immediately implies that the probability distribution $$P$$ is Gaussian. Moreover, doesn't this require that $$P$$ has mean $$m(t_k)$$ and variance $$k(t_k,t_k)$$? But wouldn't this require $$P$$ to depend on time?

Now let's say I want to model the height $$h(t)$$ of the tide at some time $$t$$. Then $$X_t$$ would represent the height at time $$t$$. The sample space $$\Omega$$ would be the real line, and $$\sigma$$-algebra $$\mathcal{F}$$ the Borel sets. But what is the probability measure $$P$$? I believe this must be a Gaussian distribution, and the kernel $$k$$ gives me the variance. But what do I take as the mean?

Your description here is accurate. I think the problem is with the notation provided in Rasmussen & Williams which I think swaps out $$t$$ for $$\mathbf{x}$$ and $$X$$ for $$f$$
Consider that your GP doesn't need to be stationary, so $$P$$ depending on time is not a problem: that is, $$X_t$$ and $$X_{t'}$$ do not need to follow the same distribution. In particular, what you say about $$m(t_k)$$ and $$k(t_k,t_k)$$ is right on, it absolutely depends on time.
For modelling tide, take a look at section 2.7 of Rasmussen. If you have prior knowledge of how the tide works, ie. its mean and variance as a function of $$t$$, then you can incorporate that into the description of a gaussian process. For instance,
$$h(t)\sim \mathcal{N}(\alpha(t), \beta(t))$$
A more familiar example for me would be $$d(t)$$, "how far am I from my bed at time $$t$$". For all $$t$$ corresponding to nighttime, $$\alpha=0$$ and the variance is very tight. For all $$t$$ corresponding to daytime, the mean and variance would change dramatically.