The willingness to hear the effect makes regret more useful than a polished account of intention.

Apologies are often prepared like statements: the reasoning laid out, the context established, the misunderstanding explained. What the other person needs is the opposite direction of travel, from you towards what they experienced. Being able to hear that without correcting it is harder than any amount of eloquence, and it is the part that actually changes something.

Next time, open with a question rather than an account. What did that do to you, and what has it been like since. Then sit with whatever comes back. The explanation can wait, and quite often it turns out not to be needed.