My questions arise on page xiv of Stix's Rational Points and Arithmetic of Fundamental Groups. Here is an excerpt:

Given a geometrically connected variety $X/k$, a fixed separable closure $\bar{k}/k$, $\bar{X} = X\times_k\bar{k}$, and $\bar{x} \in \bar{X}$ a geometric point, there is the homotopy exact sequence $$1\to\pi_1(\bar{X},\bar{x})\to\pi_1(X,\bar{x})\to \mathrm{Gal}(\bar{k}/k)\to 1$$ If $a\in X(k)$ is a rational point, there is an induced section $s_a: \mathrm{Gal}(\bar{k}/k)\to \pi_1(X,\bar{a})$, where $\bar{a}$ is a geometric point compatible with the choice of $\bar{k}/k$.

An etale path $\gamma$ from $\bar{a}$ to $\bar{x}$ on $\bar{X}$ defines an isomorphism $\gamma (-) \gamma^{-1}: \pi_1(X,\bar{a})\to\pi_1(X,\bar{x})$ compatible with the projections $\mathrm{pr}_*:\pi_1(X,\bar{x})\to \mathrm{Gal}(\bar{k}/k)$.

Then the composition $\gamma(-)\gamma^{-1}\circ s_a$ defines a section of $\mathrm{pr}_*:\pi_1(X,\bar{x})\to \mathrm{Gal}(\bar{k}/k)$, or a splitting of the above exact sequence.

Changing the etale path $\gamma$ on $\bar{X}$ varies the section over $\pi_1(\bar{X},\bar{x})$-conjugacy classes of splittings/sections. Denote the $\pi_1(\bar{X},\bar{x})$-conjugacy classes of sections of the exact sequence by $\mathscr{S}_{\pi_1(X/k)}$

Changing the base point $\bar{x}$ leads to another description of $\mathscr{S}_{\pi_1(X/k)}$ with a canonical identification between the two descriptions, which moreover satisfies the cocycle relation for composing the identifications between three choices of base points.

My questions are:

1) An etale path from $\bar{a}$ to $\bar{x}$ means an isomorphism between their fiber functors. What does it mean to be an etale path "on $\bar{X}$"?

2) Why is the isomorphism $\gamma(-)\gamma^{-1}$ compatible with the projections $\mathrm{pr}_*$?

3) Why does changing the etale path only vary the section by $\pi_1(\bar{X},\bar{x})$-conjugacy, and not $\pi_1(X,\bar{x})$-conjugacy?

4) Could someone please explain the last paragraph about the different descriptions of $\mathscr{S}_{\pi_1(X/k)}$ in a more explicit way?

Thank you.