[bmah]

On a personal level, I don't think it matters how justified your assets, personality, and talents/abilities are to dictate how you act towards others. Nor do I believe that you're falsely advertising yourself by acting modest. I believe it comes down to relating yourself with others that is the important thing here. We act modest not because we're trying to misrepresent our abilities, but because understanding, getting along, and relating with others is a social coping mechanism that most people prefer.

There are times, as stargroup has said, that you do want to highlight your abilities as much as possible, but there are still ways to do this without sounding arrogant. You can list a number of impressive things you've done without negatively implying something about the person you're telling this to. To be arrogant is IMO to be condescending, which to me often implies that said person you're talking to would not come close to achieving your level of highness. Tone of voice and wording makes a difference.

So:

1) I think arrogance is having a lack of consideration towards others. Not being able to relate to others, sense of entitlement. This is independent of how accurate your abilities are to your proclaimed bragging rights. No one cares to implicitly hear how much better you are to them.

2) There are occasions when bragging is acceptable because there are times when the context asks for it (e.g. score threads on this site). This alone I don't consider as arrogance, however.

3) If you're being modest, this isn't really dishonesty as it is a way to relate to others (again is context-dependent - oftentimes people aren't asking to show off your importance). I see what you're saying when you believe that not revealing your true abilities is a form of lying, but IMO you're interpreting the definition far too concretely.