That got corrected to "First Prince" only in later editions of EOTW. It was the only place in the whole series that "regent" was used. In the BWB, Gareth is simply First Prince of the Sword. Jordan must have realized at some point it wasn't a proper use of the term "regent".

Luc has nothing to do with anything. He was Mordrellen's son, not brother. Gareth Bryne was her First Prince.

Bryne was fully First Prince of the Sword, not First Prince regent. A regent (or a viceroy) is someone who legally assumes royal authority in the monarch's place (temporarily in absence of a ruler, or when a ruler is legally too young to exercize power. Dyelin assumed a Regency after Morgase's death and until Elayne's arrival, with the recognition of enough Houses but without the support of many others). The term doesn't apply to any other function but the royal one or its equivalent (Emperor etc.). The proper term would be Deputy First Prince or Acting First Prince, but in the BWB no such distinction is made anymore between a First Prince by birthright or appointement.

Under Andoran law, the First Prince of the Sword is the Queen's eldest brother (and legal finessing is no doubt involved that made Galad stand legally as Elayne's second brother, not her eldest one. Might be as simple as an adopted child always standing under law behind all the biological offspring, could be more involved than that. We don't know, not even if Galad is legally part of House Trakand, or adopted by Morgase but not integrated into her House. We just know it's him who stands to be Elayne's First Prince if Gawyn died. We don't know either if under law Gawyn or Galad can relinquish the position and if that lets Elayne free of appointing someone else. It sounds like Gawyn will remain legally First Prince of Sword of Andor, and in absentia he will deputize his role as Elayne's military advisor to Elayne's chosen Commander of the Queen's Guard. Birgitte, for now.). When she doesn't have a living brother, a Queen by law again is free to appoint whoever she wishes as First Prince of the Sword. Morgase first appointed her husband Taringail, and later she appointed her Commander of the Queen's Guard (he cumulated both functions), Gareth Bryne (who had been Mordrellen's, but possibly not her Commander of the Queen's Guard, unless he resigned both positions at her death, until Morgase asked him back as Commander when Lord Naldwyn died in the Aiel War. It sounds more likely Bryne resigned his position as Commander for Mordrellen when he was appointed her First Prince, which gave him authority over the Commander in time of war, then returned as Commander under Morgase after Naldwynn's death. At Taringail's death, she appointed him First Prince but unlike Mordrellen didn't replace him by someone else as Commander of the Guard).