I just got "Theory of Groups of Finite Order" by Wm S. Burnside (1897,1911).
Googling [ burnside "ac followed by ab" ] gives many links to page 3, Chapter I "On Permutations". Item 6 says "the permutation (ab) followed by (ac) .. is therefore equivalent to the permutation (abc)." I agree:
abc
bac after (ab)
bca after (ac), which is (abc)
And then "it may be similarly shewn that (ac) followed by (ab) gives (acb) as the resulting permutation." I don't agree:
abc
cba after (ac)
cab after (ab), which is also (abc) and not (acb).
Is there somebody that I can sue, or am I off base already on page 3?
There is nobody to sue, and you are indeed off base already on page 3.
The permutation you have written down is in fact $(a\ c\ b)$; you have written it in $2$-line notation as $$\begin{array}{ccc} a&b&c\\ c&a&b \end{array}$$ which shows that $a$ is sent to $c$, $c$ is sent to $b$ and $b$ is sent to $a$. So in cycle notation this is $(a\ c\ b)$. You seem to be mixing up cycle notation and $1$-line notation, which look the same but aren't.
See the Wikipedia page for more details on how each notation should be intepreted.