$$A_1=\begin{bmatrix}\lambda&1&\\&\lambda\\&&\lambda\end{bmatrix}$$ $$A_2=\begin{bmatrix}\lambda&&\\&\lambda&1\\&&\lambda\end{bmatrix}$$
These are conjugate but not similar apparently. What the?
I thought the very notion of similarity was based on being conjugate?
I.e. A_1=P^{-1}A_2P$ means that they are similar and means they are conjugate. Why are these Jordan forms apparently conjugate but not similar?
$A_1$ and $A_2$ are conjugate under the change of basis matrix $S = \left[ \begin{array}{ccc} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{array}\right]$. This, by definition, means that they are similar. You're right that conjugate and similar are synonyms in this context.