"To you" is the more common. A related common construction is "I am Liliana, or 'Lily' to my friends."
EDIT to include material from comments:
"__ to you" says "I might be called other things by other people, but this is what YOU should call me." Sometimes that could be giving special permission for someone to use a more intimate or informal address:
"Dr. Jones—" "Please, after what we've been through together, I'm 'Indy' to you."
... or it could be the opposite, a cold and abrupt way of letting someone know that they have addressed you too informally or casually:
"Hey, Mr. Jones—" "That's Dr. Jones to you."
If you want everyone to call you Lily, then "to you" isn't the best choice, since it communicates something unique to the person you're talking to. If you want most people to regularly call you Lily, then the suggestions in David Seigel's answer and its comments are best. If you rarely use "Liliana" except in legal situations and are universally called Lily, then "but I go by Lily" is great.