A one-dimensional continuous-time stochastic process is formalised as a bivariate function taking the first argument from $[0,\infty)$ (time) and the second from a probability space $\Omega$ (trajectory), or $$X:[0,\infty)\times\Omega\ni (t,\omega)\mapsto X(t,\omega)\in\Bbb R.$$ From this definition, for any $t$, $X(t,\omega)$ should be a r.v. taking $\omega\in\Omega$ as the argument. Thus the initial value $X(0,\omega)$, generally speaking, should also be random, i.e., may not be constant.
However, sometimes we do see stochastic processes with given initial value, e.g., a Brownian motion $\{B_t\}_{t\ge 0}$ with $B_0=0$, and I wonder what they mean. My understanding is that this should be similar to conditional probabilities, so under the restriction $B_0=0$, the original probability space $\Omega$ is restricted to $\Omega_0$ consisting of all trajectories starting from position $0$, and the new probability measure on $\Omega_0$ should be defined via conditional probability. Also, the original filter (if any) $\{F_t\}$ should be restricted to $\{F_t\cap \Omega_0\}$.
Am I correct so far? If yes, then, for example, how to formally define a Brownian motion $\{B_t\}$ with $B_0=0$? In particular, I'm having trouble with the conditional probability part.
Motivation: I got this confusion from reading "translation variance" of Brownian motions on page 302 of Durrett's Probability: Theory and Examples (ed 4.1), which states
If $\{B_t\}$ is a Brownian motion, then $\{B_t-B_0,t\ge 0\}$ has the same distribtion as a Brownian motion with $B_0=0$.