Apology or not?

What constitutes an apology? There are times when you hear someone say ‘sorry’ but the recipient finds the words unacceptable.
It is easy for some people to share an apology but, in truth, what are they apologising for? Lets look at a couple of examples;

1) “I am sorry that you feel upset!”
2) “I apologise for the way I acted and am sorry that I upset you?

Both the above are apologies, only one carries weight. The first apology is an empathetic one and corrects no wrong, the second accepts poor behaviour and the consequences that resulted from it. Therefore, the second carries weight and gives a greater opportunity for a positive on-going relationship.

We all make mistakes. We all have times when we must apologise.
For you, what constitutes an apology?