You’re right regarding the saber (but I don’t understand your Lucas comment)
Lucas messed up, minimizing the importance of the saber by not making it his only one.
I’m not getting nit picky though, if you want to compare the two then these things need to be recognised so you can determine which elements in each case are apples and which are oranges and then see which of them line up. Also regardless of who currently possesses it, it will always be Anakin’s lightsaber as he made and used it in its initial life. When someone legitimately claims it as theirs down the track, yes you would call it their lightsaber (ie. “Luke’s lightsabre”) but when being respectful of it’s history you would still call it “Anakin Skywalker’s lightsaber” or some descriptor as to their relation to yourself like “my dad’s lightsaber”.
Just because Disney markets the toy and replica sabers as “Rey’s Lightsaber” doesn’t mean when actually describing its history that you wouldn’t say “It’s actually Anakin Skylwalker’s second/third lightsaber which was inherited and used by his son Luke Skywalker, found by Maz Canata who gave it to Rey to return to Luke but kept it for herself without consent when Luke declined to accept it from her.”
To my point though, if you’re going to get nitpicky about the history, Luke didn’t inherit it. Obi-Wan stole it and gave it to him without Anakin’s permission. So it’s as legitimately Rey’s as it is Luke’s by your excessively literal-minded logic.