I use apologies, sorriness, and remorsefulness as three different things.
Apology is an explanation, as in Plato's The Apology. The sign in the image above that explains what happened is an apology. The usefulness of an apology is that the apologizer felt a need to give one, that is, he values the person he is apologizing to, and respects them enough to explain himself.
Sorriness is saying you feel bad that the other person was affected adversely, not that you would have done otherwise. The " This is much better" message is an example of it. The usefulness is to let the person know that you care.
Remorsefulness is (explaining that you are) feeling bad about what you did. The usefulness is to let the person know that you are humbling yourself by admitting guilt.
These three things are very important as not all situations call for all three. Indeed, sometimes using the wrong one can made a matter worse.
Apologizing is excellent when coming late (the first time), not doing things as expected, or any time the person on the receiving end feels belittled. When it is obvious that the apologizer does not care, does it quite often, or thinks apologies make everything better as if whatever-it-was never happened, the apology tends to infuriate the person.
Sorriness is to let people know you care. For example, when hearing something bad happened to someone else that you had nothing to do with "i'm sorry" means you care and means a lot to the receiver. Doing something you think is correct but adversely affects the other person, for example, a doctor taking blood or giving a shot that hurts might say "i'm sorry but this will hurt". In other words, i do not regret the action, but i feel bad that it will hurt you.
Remorsefulness is expected when people have their values slighted. A convict may be given a more lenient sentence if he shows remorse, a child may be forgiven if he really does feel bad that he broke the friend's toy, a relationship can be salvaged this way as well. Remorsefulness is the hardest of the bunch because the person must humble himself, a quality not often found in society.
The method of remorse is really based on the relationship the two parties have. The idea is not the words but the conveyance of actual humility. As this is based on many factors including the people involved, the amount of time taken between the offensive action and the expression of remorse, and the gravity of the offense, no real guidelines can be explained. Some can be via email, some cannot, though perhaps it can be a first step.