I'd say that there are many "errors" that could betray a "synced" video, but it could be theoretically possible to avoid all of them, so there's no way to guarantee authenticity (short of reliable and trustworthy witnesses, etc.). Some telltales:
- The easiest might be if a shot shows an instrument being played and you know the technique is wrong. The other day on a TV show a cellist was supposed to be playing a rapid passage and the actor wasn't even moving their left hand. Often, especially with bigger budgets, actors might put in more effort, but just as with lip syncing, you might notice timing mismatches or simply a mismatch between the level of exertion and the resulting sound. Of course, a video that carefully avoids showing the instrumental technique can be suspicious. E.g. it might show an actor playing piano, in a shot that doesn't show their hands, and then intercut shots of a real pianist's hands but without showing the face.
- This can be tricky, but mismatch between the visible micing and the sound. E.g., if I'm playing a lute in a cathedral, I'll probably want at least one mic to be within 20 feet of me, perhaps ideally an array of several close mics plus some farther away for room ambience. A "long shot" showing the instrument with no mics in sight for many meters might be a giveaway. This can be tricky, as I said, because there could be pickups or well-hidden mini-mics and wireless systems, but an experienced audio engineer might be able to spot a difference between the sound and the visible means of capturing it, just as an instrumentalist can spot fake technique. (By the way, this can be one very good reason for "syncing" a performance. Often the best sound and the best shot require the mic to be much closer than the camera, and perhaps the mic arrays would be "cluttering" up the shot undesirably (if you've got a bunch of mics and stands in front of the lutenist's instrument and body). Making my own quick-and-dirty videos, I often leave the mic and mobile recording device in the shot, run the camera simultaneously, and then sync the audio to the video afterwards.)
- This one as well can be tricky, but a mismatch in room ambience. E.g., if a video is shot outdoors, in a meadow, there will be zero reverb naturally. Now, it would be possible to mic very closely and then add artificial reverb, so depending on how "produced" the sound is, this could be hard to quantify. But it would probably be easier, with any location that isn't as quiet as a recording studio, to record elsewhere and sync. Watching the "making of" videos of movies, it's surprising just how often movies will film on location, and then entirely dub over the dialogue in the studio (especially for action scenes where stunt equipment, wind machines, etc. might ruin the audio).
- Similar is simply "improbable recording challenges." I'm no audio engineer, but looking at the Jinjer Pisces video, they're all on top of each other, with the vocalist right next to the drums. There would surely be significant bleed from the drums into the vocal mic. As I say, for all I know, it could be possible to capture this well between mic directionality and filtering, but it would certainly be easier to just put the vocalist in a booth in the studio. Similarly, one reason that superstar performers (like Whitney Houston) might lip-sync their stadium concerts is not just to ensure a good performance even if they're having an "off day," but simply that the sound engineering is a lot easier if you only have to worry about amplification than if you also have to worry about capturing the sound in a difficult environment.
Now, note, there could be such a thing as a "mostly authentic" live video. E.g. what if I record three takes, all live. In editing the video, I like the audio of one take best, but I want to include a camera angle for a few seconds that was captured in another take. For those few seconds, there might be a mismatch in synchronization, but it doesn't mean the whole video is "fake."
All this means, if it's important to guarantee authenticity, to minimize the things that might give away fakes. E.g. if you're recording a video to audition for a competition, music school, etc. Record in one camera take, with no changes of angle. Position the camera both to show faces and the operation of any instruments. Don't worry about keeping mics out of the shot. If you do anything "in post," like syncing captured audio to video, do everything you can to ensure they sync well (this can be harder than it sounds!).
And if you're on the receiving end of such a video, I suppose someone could take all those steps and still "sync" their video. If for some reason it's vital to tell the difference, maybe the only way to guarantee authenticity is to have the recording vouched for by someone you trust. But at some level it doesn't matter; if the purpose of the video is to prove that the performer can do what you hear, it could be very hard (at least for an instrumentalist) to fake something they're not capable of. I could "sync" video to my own audio, but I couldn't make a convincing fake to Itzhak Perlman's recording of Paganini's fifth violin caprice, because I simply can't replicate his performance. You don't have to feel too "cheated" by the two videos you list: Everybody knows Whitney's got pipes; it's not like the Monkees syncing their moves to other people's talent; that's really her singing, whether it's right that second or not. Similarly, the Jinjer Pisces video does a very thorough and accurate job with their syncing; the guitarist and drummer are clearly really playing the same thing that you hear (as far as I can tell not being a guitarist or drummer). Any difference between what you hear and what was produced during the shoot would likely be in audio quality, not performance quality.