Hey all! First time poster - long time reader (love this forum!)
Wanted to address question of distinction between "peers" and "commoners."
I do not believe that Harry is technically or legally a commoner - and would specifically the various royal patents (most importantly Geroge V's in 1917 conferring the title HRH and all honors on male-line grandchildren of the Sovereign).
All peerages derive from royal patents (Sovereigh - font of all honours - whether bestowed in 1400 or 1917).
This is further reflected in Harry's role as Counsellor of State, and the fact that his passport would identify his as HRH Prince Henry of (Wales? the United Kingdom? whatevs) as opposed to Henry Mountbatten-Windsor.
The distinction between peers and commononers (ie: seat in the House of Lords) in British common law does not neatly correlate to identification within the Royal Family.
Bottom line: legally Harry, Beatrice et al are pears of the Realm by virtue of the 1917 Letters Patent - even though they are not peers in the House of Lords.
My two cents.
No, no, no. You misunderstand what these letters are designating.
Luvcharles's explanation is correct. The letters you are quoting set out who can and cannot be a prince and be an HRH. Nothing to do with peerages or commoners.
During George V's time, there were lots of HRH and princ(es) of the realm by virtue of being related to the royal family eg Fat Mary's daughter Mary who later married George V was a princess despite being Victoria's second cousin once removed!!
Those letters patent were George's version of streamlining the royals like Charles is doing. He therefore restricted who could be an HRH and who could be a Prince. The letters restrict them to child/grandchild of the monarch in the male line and no one else in the family even if they are blood. This is why you don't see the kids of Gloucesters, Kents with the style of Prince which they would have been able to call themselves prior to the letters being written.
This restriction had to be extended by Lizzy for PGtips + siblings because it did not cover GREAT grandchildren. If she hadn't extended it, PGtips would be Lord George Of Cambridge.
The same reason is why Gloucesters and Kents are Princes, but their kids are not. The Gloucesters are George V's grandchildren so they are all Princes ie Princess Alexandra, Prince Richard (Duke of Gloucester), Prince Michael and Prince Edward (Duke of Kent)
As the letters set out this designation in the male line, no female can extend this to their descendants unless the Monarch specifically decides to grant it. Therefore Anne's kids have no titles and aren't HRH or Princes because she can't pass them on, being female.
Further, the Queen's own father had to write a special dispensation to make *her* kids HRH + Princes despite the fact that Lizzy was going to be the monarch eventually.
Edward Wessex's kids may go by his lesser titles, but as grandkids of the monarch, they are automatically and legally HRH + Princes.
Beatrice, Eugenie, James, Louise, William and Harry, being grandchildren of the monarch, descended from the MALE line, are automatically HRH + princes.
Finally, being a peer of the realm excludes the ruling family. Until the recent reforms, being a peer gave one automatic seat in the house of Lords.
As LuvCharles has explained, Harry, Michael, Beatrice, Eugenie, Anne aren't peers and therefore can't take up seats in the house.
As for Harry's passport, his style is HRH + Prince in the same way that one would be Mr or Miss or Mrs or Ms. His name is Henry. Since he has no last name, he adds his father's dominion house ie 'of Wales'. Mountbatten - Windsor isn't a surname. It's a dynasty name. For the purposes of legal documents where a dominion house designation won't do eg when William sued the french paps, then they use Mountbatten - Windsor, but otherwise they all use their dominion name. Eg Charles's kids are 'of Wales' or Harry Wales for short. And Andy's kids are 'of York' or Beatrice York for Short. And Edward's kids prefer Windsor, but we don't know yet whether that will stay when/if they are elevated to 'of Edinburgh'