It seems we get one from any symmetry of the diagram; are these all of them?

No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.

Edit: The outer automorphism group of the Coxeter group $E_8$ appears to have order $2$ and is generated by $w \mapsto w_0^{\ell(w)} w$ where $w_0$ is the longest element, which is central. I found this result in this thesis by William Franszen.

It seems we get one from any symmetry of the diagram; are these all of them?

No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.

Edit: The outer automorphism group of the Coxeter group $E_8$ appears to have order $2$ and is generated by $w \mapsto w_0^{\ell(w)} w$ where $w_0$ is the longest element, which is central. I found this result in this thesis by William Franszen (which has much more information about this problem), which was the second search result I got for "outer automorphisms of coxeter groups."

It seems we get one from any symmetry of the diagram; are these all of them?

No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.

It seems we get one from any symmetry of the diagram; are these all of them?

No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.

Edit: The outer automorphism group of the Coxeter group $E_8$ appears to have order $2$ and is generated by $w \mapsto w_0^{\ell(w)} w$ where $w_0$ is the longest element, which is central. I found this result in this thesis by William Franszen.