Is acting 2-transitively stronger or weaker than acting transitively?
I'm struggling to get to grips with the definition.
At first I thought it was a weaker condition: If the graph of a "transitive" action is connected, then I thought acting 2-transitively would mean the graph consisted of a forest of two trees. But I can't square that up with the definition. I think it must mean something different, in fact something stronger rather than weaker.
I grasp ideas better with simple illustrative examples, onto which I can pin definitions, rather than definitions alone.
Being 2-transitive is a stronger (more restrictive) assumption than being transitive. Here are two examples of groups of transformations on the real line.
The group of translations of the real line is transitive. Given any two points, you can translate the real line so that one point is moved to the other. However, it is not 2-transitive. If we have, say, the pair $(1, 2)$ and the pair $(2, 4)$, there is no translation that moves the first pair to the second pair.
Now take the group of translations, non-zero scalings, and reflections of the real number line. It is obviously still transitive. However, this time it is also 2-transitive. Indeed, we can transform the real number line under this group so that $(1, 2)$ is moved to $(2, 4)$. And the same goes for any other two pairs of distinct real numbers. (The pairs are ordered, which is why it's important that this transformation group also has reflections / negative scalings.)
If a group acts 2-transitively on a set with more than one element (what it means for a set with a single element is a matter of definition and convention, not of proof), then it always acts transitively as well: Take two points $x$ and $y$. If they are equal, use the identity action. If they are not equal, then you can move the pair $(x, y)$ to the pair $(y, x)$ by 2-transitivity. This has moved $x$ to $y$, which is what we need for transitivity.