...She wasn't sure what to say to that. She really was an unimportant chuunin. Her mission wasn't high priority or covert, so knowing her name would mean nothing.

Unless she started the Fourth Ninja War by falling into Ame, then maybe her name would be important. There was that.

"...Sakura. Sakura Haruno, of Konoha." Sakura was the name she defaulted to, because there was no use in being identified by a family that meant nothing even inside Konoha. But presuming that a lord-would-be-god-representative would want to use her first name would not win her any 'not going to start a war' points. Probably.