Ish, I like that, that the Queen reigns because the parliament asks her to. This could be compared to the situation in medieval Scotland, where the kings were elected by the parliaments of the landed gentry--the common people had no input, from what I read. In England today, supposedly the parliaments are elected by the common people, so therefore the Queen reigns by permission of the people. We all know, of course, that selfish interest governs political bodies in all centuries, and parliament is no more to be trusted than imposition of rule by warfare, but at least the situation is coherent with a moral outline of republicanism.
My Scots ancestors removed from England to Scotland at the behest of David I of Scotland, along with others who supported Queen Matilda over King Stephen. For this reason alone I would be pleased with the naming of the heir Matilda, if it is a girl. Perhaps David would be an equally interesting name for an heir...but that's argued endlessly on another thread.
The English monarch does rule by the permission of Parliament - we've seen this increasingly in British history since the reign of Charles I - the whole Civil War can be seen as a fight between Parliamentarians and Royalists over who ruled with the permission of the other, and the Royalists lost. The government deposed Charles I, the government invited Charles II to be king, then deposed James II and invited William and Mary to be co-monarchs. The Hanovers came to power because the government selected them as the best heirs, the Abdication Crisis happened because the government did not want Edward VIII's chosen spouse.
The early medieval Scottish and Anglo-Saxon practice of having a collective of lords who advised the king and had the power to select or depose of the king is a big part of why the English/British parliament has this power while other European governments historically haven't. Instead of relying on pure primogeniture they looked at the royal house and at times selected the best for the role. It's how Alfred the Great became king despite the fact that his elder brother (who was king before him) had sons.
We can argue that this is what happened with Matilda. While the court had sworn to uphold her as the heir of Henry I, when he actually died and Stephen appeared to claim the throne they went with it. Matilda eventually fought back, and for a brief period in 1141 she ruled. As for the support she received from David I of Scotland, it makes a lot of sense. David's sister, Matilda of Scotland, was Matilda's mother. David had enjoyed the power and status in England that came from being the King's brother-in-law, he likely wanted to continue it by being the Queen's uncle (similarly, Leopoldo I of Belgium tried to have that kind of influence over his niece, Queen Victoria).
Jane's brief rule can be seen as an attempt to disrupt the legal succession by the permission of the government - it only fails because she didn't have the huge support that Mary gathered. It almost makes you wonder if Elizabeth had attempted to usurp her sister in Jane's place, would she have gathered more support, or would she have ended up executed instead? From the perspective of Mary's Catholic monarchy, executing Elizabeth would have been advantageous - she could have then appointed Mary of Scots as her heir.
No-one really refers to Bonnie Prince Charlie as Charles III so it's not that relevant, but Prince Charles will be Charles III, assuming he's abandoned the idea of calling himself George VII.
Jacobites refer to him as Charles III, much like how they refer to the current heir, Franz, Duke of Bavaria, as Francis II - if a Jacobite revolution took place and Franz was put on the British (or an independent Scottish or Irish) throne, that would be his numbering due to the Jacobite line.
The Jacobite line doesn't pertain to the Windsor line, however. Charles will be Charles III (if he goes with that) because the two lines have been diverted since the Glorious Revolution.
History books always refer to Jane as "Lady Jane Grey", which is totally inaccurate - even if you don't want to call her "Queen Jane", she should really be referred to by her married name of Lady Jane Dudley, which I think was how she was referred to during her trial.
It's not inaccurate to call her Lady Jane Grey. Historically, women are often referred to by their maiden names (not universally so, but often). Hence why we remember Queen Mary as Mary of Teck, or her mother-in-law as Alexandra of Denmark. I don't know if she was referred to as Grey or Dudley in her trial, but I do know that in Fox's Book of Martyrs (first published 10 years after Jane's execution) refers to her as Lady Jane Grey.
The parallel's not usually drawn because the Tudor claim to the throne was so tenuous, but in some ways Henry VII was in the same position as Henry II - his claim came through his mother, and she was still alive when he became king.
The Tudor claim to the throne was not actually tenuous. Henry VII was descended from the John of Gaunt through the Beaufort line. It's a tenuous claim in itself, yes, except for the fact that Henry was out forth as a possible Lancaster heir during the reign of Henry VI, and was the most senior (in terms of descent) man in the Beaufort/Lancaster line following the decimation of the Lancasters in Edward IV's second reign. Henry VII's claim was then solidified through the act of conquest, and further solidified through marrying Elizabeth of York, the eldest daughter (and remaining heir) of Edward IV - all acts that are comparable to the rise of Henry I when you think of it.
That Henry VII inherited while his mother was alive is seen as a sign of the idea that women couldn't inherit. This is what plagued Matilda's life - she was supposed to inherit, the Barons swore during her father's life that she would inherit, but when push came to shove they chose to instead support the closest male claimant - Stephen, who also came to the throne while his mother, Adela of Normandy, the eldest daughter of William I, was still alive. The idea here is that women could not inherit the throne - and they did not until Mary successfully inherited it.
In the case of Matilda, however, she didn't simply inherit the throne - she also briefly succeeded in seizing it. My point initially is that while Matilda undeniably held the throne for 7 months, longer than Edward V, her reign is disputed as being such, while his reign is not. My reasoning is that since we haven't had a Matilda II, we can't say for certain that we've had a Matilda I (or, since no one since then has been Matilda I, we can't say Matilda wasn't the first). We donn't dispute Edward V's claim because when Edward Tudor came to the throne it was as Edward VI, thus retroactively confirming the short reign of Edward V as valid.