> To be an Empire, wouldn't Israel need to defacto control _some territory_ and then be trying to expand their empire into Palestine?

They very openly do both those things in the West Bank.

Then you have the more extreme settler types talking about the biblical "Land of Israel" that would extend into modern Egypt, Syria and Lebanon.

But as far as I understand it, Israel is usually not the empire itself, but a bridgehead or particularly glaring example of imperialism from the West, starting with the British Empire.

> Or is the idea that the US is the Empire?

So, yes.

It were those countries that conquered those regions from the Ottoman empire and then decided among themselves to support the project of a Jewish state, against the wishes of the existing population of the region.