I've been trying to do research on this subject and it has been so frustrating! You can find dozens of pages describing how titles worked in the 17th and 18th centuries, but virtually nothing on how titles worked in pre-renaissance (medieval) england, france, germany, and italy. With that in mind, here's what I want to know.
I know these questions may sound trivial, but I'm trying to create an accurate titles system for a video game mod I'm working on.
Ha, I've just put together an answer to another question which happens to include some stuff that's relevant here. In fifteenth-century England, anyway, the son of a king was definitely a prince. Partly borrowed from my other answer:
In 1454, King Henry VI was suffering from some form of mental illness, to the point that he was unaware of the birth of his son. Here's John Stodeley's newsletter from that January, a few months after the child's birth:
At the Princes comyng to Wyndesore, the Duc of Buk’ toke hym in his armes and presented hym to the Kyng in godely wise, besechyng the Kyng to blisse hym; and the Kyng yave no maner answere. Natheless the Duk abode stille with the Prince by the Kyng; and whan he coude no maner answere have, the Queene come in, and toke the Prince in hir armes and presented hym in like forme as the Duke had done, desiryng that he shuld blisse it; but alle their labour was in veyne, for they departed thens without any answere or countenaunce savyng only that ones he loked on the Prince and caste doune his eyene ayen, without any more.
(When the Prince came to Windsor, the Duke of Buckingham took him in his arms and presented him to the King politely, asking the King to bless him; and the King didn't respond at all. The Duke stayed with the Prince beside the King anyway; and when he couldn't get any kind of response, the Queen came in, and took the Prince in her arms and presented him the same way the Duke had, asking the King to bless him; but all their labour was in vain, for they departed from there without any answer or recognition except that once he looked at the Prince and then cast down his eyes again, without any more.)
And it's not just John Stodeley using that term for the child. When Henry recovered, a year later, Edmund Clere wrote to John Paston:
Blessed be God, the Kyng is wel amended, and hath ben syn Cristemesday... And on the Moneday after noon the Queen came to him, and brought my Lord Prynce with her. And then he askid what the Princes name was, and the Queen told him Edward; and than he hild up his hands and thankid God therof.
(Thank God, the King is much better, and has been since Christmas Day...And on the Monday afternoon the Queen came to him, and brought the Prince with her. And then he asked what the Prince's name was, and the Queen told him Edward; and then he held up his hands and thanked God.)
And it wasn't just a general form of reference to a noble child, it was the child's official title. There was a ceremony where the prince was 'created' (had the title officially bestowed on him). Botoner to John Paston, 8 June 1454:
as to nouveltees, &c., the Prince shall be create at Wyndesour, uppon Pentecost Sonday, the Chaunceller, the Duc of Bokyngham, and manye othyre Lordys off astate, present wyth the Quene.
(As for news, the Prince shall be created at Windsor on Pentecost Sunday, with the Chancellor, the Duke of Buckingham, and many other Lords of estate present with the Queen.)