That the mistake is made again does not in and of itself mean that the apology was insincere. Not everyone can learn from their first mistake.

Anyway, great messges throughout your essay. There is definitely a right and a wrong way to apologize and for an aplogy to really mean something it should be much more than an apology, it should be an amends.

In the latter third of my essay, https://medium.com/know-thyself-heal-thyself/spirituality-redefined-ff45cb53eaf2 , subtitled, "plus a program for living a spiritual life," I talk about the spiritual wisdom and value of steps 9 and 10 in any 12 step program (9 is making amends to all we have harmed in the past unless to do so would cause more harm and 10 is continuing to take a personal inventory and promptly admitting to new wrongs) and talk at length about the right way to apologize/make amends, for example "Be humble and don’t assume anything, Accept responsibility for actions and what those actions brought, Admit fault, Change behavior and demonstrate that healing has started, and Make an effort to rebuild trust."

And making proper amends helps both parties:

"Because making amends is like relieving yourself of a burden, a burden of knowing that you caused someone harm and failed to do anything about it. You stood by while the other person suffered in some way, and for many people who do this, their own actions often cause feelings of guilt, shame, anger, self-hatred, and depression. When we offer amends, there is often an unburdening of those feelings. It is like a spiritual detoxification process. So in addition to making an offer to help relieve the other person of suffering, making amends can help you feel better, too"