Latin phrase for Roman Army?

[Nathan Ross]

I am not at all sure that the Romans used the word imperium in the same sense that we use 'Empire'.

That was my first thought too. However, it does seem that the usage developed over time. This is one of those interesting-looking books I've long been meaning to read - the chapter here has some discussion of a later Roman 'shift in meaning' of the word imperium to denote 'a bounded territory of empire' rather than a sphere of authority:

News and Frontier Consciousness in the Late Roman Empire

Respublica, meanwhile, I've always taken to refer to the Roman state generally, the 'public affairs', rather to a specific style of government or geographical area of control.