I was thinking about a question on here earlier, and came up with this question.
[Added Hausdorff note, below.]
It is easy to see that the group of self-homeomorphisms of the real line acts $2$-transitively on the space, but not $3$-transitively.
Likewise, it is clear that, for example, $\mathbb R\setminus \{0\}$ is a space on which the group of self-homemorphisms is $1$-transitive but not $2$-transitive.
If you take the real line with the open sets of the form $(-\infty,a)$ for some $a$, then I guess this gives you an example, but that space is not Hausdorff.
I can't seem to think of an example of a connected Hausdorff space where the self-homeomorphisms are $1$-transitive but not $2$-transitive.