“My bad” covers stepping on a shoe. It doesn’t cover breaking a promise.

Proportion is what makes an apology believable. Two casual words are exactly right for an accident that cost nothing, and those same two words aimed at something that genuinely hurt somebody tell them you have not understood what happened. The small apology on the big thing reads as wanting to be finished with it.

Match the repair to the damage the next time you owe one. For something that mattered, say what you did, say you were wrong, and say what you will do differently. Then let them react for as long as they need to.