When my son was small and was asked to apologize to his sister for some long forgotten transgression, he sighed deeply, scrunched up his face, turned red and blurted out “Sorry GaGa.” Now his sister’s name isn’t GaGa, that was just his way of mitigating the apology.
If you listen closely to many people’s apologies, they’ll say something like ” I’m sorry IF I offended you.” That’s the adult way of saying “Sorry GaGa.” The ‘If’ turns an outright apology to a conditional one and therein lies the problem. The apologizer (is this a word) is failing to accept full responsibility for his actions. (Please note that I have used the masculine form of his and hers because it is usually men who are wrong and who need to apologize.)
What we want is for an apologizer to acknowledge fault and fully accept responsibility. Accepting full responsibility defuses any situation immediately, takes the wind out of the sails of of the complainant.
The best thing to do is to live by the motto “If in doubt, apologize.” If you have the slightest concern that you may have something worthy of an apology then you probably do. You can even go one better and apologize ahead of time, knowing that you might need to build up apology credits to get out of the doghouse at some future date.