Performing for a large audience is a technical challenge in terms of getting the best quality sound to the entire crowd. So if a performer wants to lip sync the song that they themselves actually prerecorded that is their right. With this, however, they will have to suffer the consequences if the performance goes wrong. The singer is only a true fake if they didn't ever sing the song.
No, lip syncing at live events does not mean the performer is a fake. There are many reasons why a performer may need to lip sync during a performance. If the artist has a cold or another ailment that may affect her voice and she decides to lip sync, this does not make her a fake.
Lip syncing, even at a live event, is not a bad reflection on the singer, and the performance is not fake. As seen on the live broadcast on New Year's Eve, it is not easy to do. It still takes a lot of rehearsal and coordination with the production team.
No, lip syncing at a performance doesn't mean that a performance is fake, because there are a lot of things that go into a performance besides singing. These days, a singing performance is as much about showmanship as it is about singing. On New Year's Eve, the temperatures were very cold and Mariah Carey was in costume. It shouldn't have mattered that she wasn't singing.