The one case (for $\frak g$ simple) where the two definitions of real forms (up to $Aut(\frak g)$ versus $Int(\frak g)$) don't agree is the following.
There is a real form of $\frak g=\frak s\mathfrak o(2n,\mathbb C)$ denoted $\frak s\frak o^*(2n)$.
For $n\ge4$ even $\frak g$ has two subalgebras isomorphic to $\frak s\frak o^*(2n)$ which are related by $Aut(\frak g)$, but not by $Int(\frak g)$. If $n\ge 3$ is odd there is only one such algebra up to $Int(\frak g)$. The difference is because the center of the simply connected group is $\mathbb Z/2\times\mathbb Z/2$ if $n$ is even, and $\mathbb Z/4$ if $n$ is odd.
$n=4$ is particular particularly interesting: the triality automorphism of $D_4$ interchanges these two copies of $\mathfrak s\mathfrak o*(2n)$, as well as $\frak s\frak o(6,2)\simeq \frak s\frak o^*(2n)$.
