Your understanding of British titles is rather inaccurate.
There is no such title as Prince of Edinburgh, nor Princess of Edinburgh. Likewise, there is no Prince/Princess of York, Prince/Princess of Gloucester, Prince/Princess of Kent, Prince of Cambridge and so on.
British Royals are Princes/Princesses of the United Kingdom, nothing more. The heir apparent may also hold the titles Prince of Scotland and Prince of Wales (Prince of Wales is not an automatic title though), but other than the heir apparent no British Prince or Princess holds any title that makes them a "Prince of Location" other than the UK as a whole.
It is custom for the children of British Royal Dukes to use their father's title as a territorial designation - hence, the children of the Duke of Edinburgh use "of Edinburgh" as their territorial designation, and so on with the children of the Prince of Wales, Duke of York, Duke of Gloucester, Duke of Kent, Duke of Cambridge, and so on. The children of the monarch use no territorial designation whatsoever, and territorial designations are dropped when further titles are acquired.
Thus, Charles and Anne were born HRH Prince Charles of Edinburgh and HRH Princess Anne of Edinburgh, as the children of the Duke and Duchess of Edinburgh, but ceased to be "of Edinburgh" when they became the children of the monarch; Andrew and Edward were never "of Edinburgh" as they were born children of the monarch.
William was born "of Wales" as a child of the Prince of Wales, but ceased to be "of Wales" when he became Duke of Cambridge. Harry will likely cease to be "of Wales" when he marries, as he will likely receive his own dukedom then. His children will never be "of Wales". They will not be born Princes/Princesses of the United Kingdom if they are born while the Queen is alive - the LPs in existence right now only allow for the children and male-line grandchildren of the monarch, and the children of the eldest son of the Prince of Wales to be HRHs; Harry's children will only be HRHs once Charles is King.