As with most ancient societies and even more advanced ancient civilizations, the spiritual needs of ancient Rome were met with religion. A large number of pagan gods were worshipped in ancient Rome. In many cases the Roman government had basic pagan gods and rituals that were followed in order that the gods would be pleased with Rome's endeavours.

For government "sponsored" gods, there was the Imperial Cult as an example of a "centralized" religion. The imperial cult cut across all the lesser gods and cults. This was a unifying force that helped hold Roman society together. The Roman State appointed special priests and built special temples for the highest level gods of ancient Rome.

Severus

Mithraism was not an imperial cult, it was a military one. It thrived under several emperors. We know little about it as its rites were secret. We do know that it originated in ancient Persia and that the god Mithras was always seen killing a bull.

It's a matter of words and how they are used. The city of Rome is the city itself. Imperial Rome is often used to designate Rome's authority. Such as Dacia became a part of imperial Rome, or Pompeii was a city in imperial Rome. Imperial Rome can be used as a synonym for Roman empire.

The ancient cult met in secret.