"I'm. Mo Xuanyu," he says. Not true, but it's the closest he can get without having to explain a number of things.
"...I don't have a family."
He has no intention of going back to Mo Village and pretending to be Madam Mo's nephew. Mo Xuanyu might have had similarities to Wei Wuxian, but they weren't the same people and this would become obvious in time. He also doubts his chances of being taken in by this body's half-brother in Carp Tower. Jin Zixuan had managed to pull himself out of being a complete asshole by the time he died, but this would be the stuck up brat that he was before the Sunshot Campaign.
As for Lotus Pier - he very desperately wants to see the people there, but right now they have no reason to take him in. Uncle Jiang had only brought Wei Wuxian into the family due to his deep friendship with Wei Changze.
It does occur to him that claiming the name of Mo Xuanyu instead of making one up means he might wind up getting sent back to Mo Village. As long as he manages to sneak away from Lan Zhan long enough to check on his siblings and his past self he won't put up a fuss.