Basically, dehydrated salamanders don’t run away from a poke in the butt as fast but they do move faster when searching for water. Is that it?

Yes, exactly, that's it!

When you poke the salamanders, the hydrated salamanders move faster than the dehydrated salamanders.

The dehydrated salamanders are more motivated to find water. They find the water faster because they actively try to seek water.

From the article:

we consider that the difference between these two results may be attributed to the fact that dehydrated salamanders could be more motivated to search for a water source than those hydrated. On the contrary, in the locomotor performance test, as all salamanders were stimulated or forced to move, it is probable that hydrated animals were able to perform a higher speed than those dehydrated due to their better physiological condition.