Losing Face is not only to suffer from embarrassment, but also usually due to something that you did that goes against what you really are like, what your natural character is. If you promise to do something and then break that promise, you'd lose face to the person you made the promise to (and anyone witnessing this). The term applies to not only doing something that is against your character, but also is witnessed by others. The idea is that your face is a characteristic of yourself that others see. So by losing face, you are showing others something that is beneath your normal moral character. So if you tell a lie but no one catches you in that lie, it isn't really losing face since no one witnessed your fall in character. Likewise if you trip and fall and someone sees you doing this, you aren't losing face since everybody has accidents and this won't tarnish your character (unless you are supposed to be steady on your feet at that particular moment as in a ballerina that trips during a performance).
Note that you can do something dishonorable that is witnessed by others, and it may not be considered losing face if your normal character is dishonorable in that light. A fine-art forger wouldn't lose face by selling a fraudulent copy of some Rembrandt. But that same art forger would lose face if they were caught selling the fake painting because their fake was easily spotted as being a forgery. A burglar breaking into a house and stealing the person's stash of diamond wouldn't lose face for that act, but may lose face if the burglar had to kill the occupant in the process. It's all relative to that person's moral compass.
Also note that someone doesn't have to have a first-had witness to the act to lose face; a person admitting to their mistake and bringing shame on themselves by admitting to that fault can still lose face to those (s)he told.
Finally a person may lose face to some witnesses and not to others for the same incident, depending on the perspective of the witnesses. A drug dealer that tells the police about a major drug shipment would lose face with other drug dealers, but may actually gain face with the non-crimial part of society. This is because to drug dealers squealing to the police is considered "bad" in their world whereas law-abiding citizens would think that it is "good".