I need to make one small correction to your question. There are several misconceptions about Latin and Roman rule. One is that One is that the Latin 'imperium' means empire and the other is that the Roman emperor was called 'imperator'. The title 'imperator' was honorific and meant that he had been hailed 'imperator' by his troops at the end of some great victory. 'imperium' was the Latin for the sphere of command allocated to a senator after his year as consul, or quaestor. The term 'imperium' meant 'sphere of command'. So, after his year as consul, as ex-consul would be given a provincia, and his imperium to raise troops as necessary, levy taxes, keep the peace held exclusively within the boundaries of the provincia. Inside it, he had the imperium (power), but outside it, he had no power at all. He was just an ordinary citizen. We owe the use of the word 'imperalistic' word imperium to the Roman poet Virgil who has Jupiter promising that Rome (meaning his Roman readers to think, of course, of the then current emperor, Augustus Caesar) was destined to have fame and 'imperium sine fine' - unbounded imperium. Each individual province of each provincial governor had a defined finis (boundary). Augustus' authority had none.
You are right, though, that the Roman 'emperor' had the title 'princeps' and that this reflected the fact that the he had the permanent title of princeps senatus (head of the Senate). Roman historians of the first century CE, like Tacitus, regularly referred to the 'emperor' as princeps. But the princeps was never hereditary, even when it was temporarily dynastic. The emperor of the day decided who should follow after his death and at some stage gave him and associate in his powers, so that, when the emperor died, the successor was already in power. So there was never any need for a special title for 'heir apparent'.
The nearest Romance language to ancient Latin is Italian. And Italy by the early middle ages had fragmented back into a number of city states, not necessarily with any strict constitutional arrangements. The Italian word principe actually referred in Sicily to its ruler. The ruler of Venice was a Doge. But the word for king that prevailed in Italian became the Italian word re for the original Latin word for that role, rex.
William I gave his previous title Duke of Normandy to his William Rufus, his son and heir to the throne of England, later to be known as William II. The practice of calling all male offspring of the king Prince began, according to Etymonline, in the sixteenth century.