I don't know if you already got your answer, but I grew up in a Western Monarchy (Denmark) and there the crownprince/ss is defined as the next in line for the throne. So when Queen Margrethe II was the monarch (in Danish the word "regent" is used also for when a person is ruling in their own right, so I'll use monarch for the most part) and her oldest son, now King Frederik X, was Crownprince Frederik. His wife, now Queen (Consort) Mary became Crownprincess at their wedding in 2004. Their oldest child, now Crownprince Christian was "simply" Prince Christian until his father assumed the throne in 2024.
Others have mentioned this already, but since men in monarchies historically and (somewhat) legally(? for lack of a better word) have more power than women, it also meant that Queen Margrethe's late husband, Prince Henri, could never assume the title of King, since a king holds more power than a queen, and because he was never in line for the Danish throne, he was always going to be under his wife, and therefore he was also never a Crownprince.
I couldn't find any info on what Margrethe II's official title was before she was queen (she's been long before I was born), but she only became next in line for the throne in 1953, after a change in the The Constitution of Denmark, which allowed for women to became Monarchs in case there was no male heir (Her father only had daughters). The Constitution was further changed years later to reflect that any firstborn of the Monarch, no matter the gender, could inherit the throne.
As for prince/ss regents, in Denmark there are two types of "regents" in Denmark (here using OPs definition since again in Danish "regent" is used as a synonym for monarch) for when the current monarch is for any reason unable to do their job. The difference in the two, I think lies both in the title and the role. For instance when King Frederik X is for any reason unable to do his "kinging" work, Crownprince Christian's role is being the regent (Danish definition)/monarch. For the other three who are able to "rule" when the King (and Crownprince) are absent, their title, I think more translates to "Head of State". The King's aunt Princess Benedikte and his younger brother Prince Joachim are both able to be Head of State whenever the King and Crownprince is absent. Also a huge change was made a few years ago, when Queen Mary was made Head of State, despite not being in (blood)line for the throne. I think also Queen Margrethe is a Head of State, but I'm not too sure (I'm using the official website for the Danish Royal family as my source)
Of course I realize you probably didn't feel the need to know all this, and this is very specific to Denmark, but I am a royalist by heart who talks too much.
bad tldr: crownprince/ss = going to be king/queen at current monarch's death/abdication. Prince/ss regent/head of state = doing current monarch's job whenever they are temporarily absent.
Also this is probably different in other Western monarchies, my interest purely lies with the Danish one