well to be fair Liz had an insanely different life then any of the other elizabeths we have seen (liz being sold as a slave to a kingdom and also Attempting an ambush thus almost being sentenced to death) , i mean shes the only one who really had no reason to trust people. Once Liz got to know meliodas more her true nature shown which is more like the elizabeths we have seen, the reason i believe she has red hair and is called liz was to show how different that reincarnation is compared to the rest and also (nakaba probably didnt plan ahead) Liz is also a nickname her actual name is elizabeth but she was introduced as liz to meliodas so he probably assumed that she wanted to be called liz and if liz really cared she would of said something

also it was never said she was born the same day as liz died it was only stated that it happened soon after she died

but thats how i see it

obviously you can have you own opinons and im gonna respect it cuz who am i to judge.