That's not a bad explanation.
Anywho, time to play devil's advocate for the sake of discourse.
From what you have said, one can be led to believe that morality is inherently greedy, and by saying that helping another because it gives us satisfaction is moral, we agree that the fulfilled sum of our desires is completely moral. If it is moral to act in a manner that satisfies us in such a situation, is it then moral to act as a hedonist or glutton?
What you have described at the end of your first paragraph is that you must help people for their own sake, rather than your own pleasure, in order to not be selfish. You have contradicted yourself, though, and said that people lie when they help others selflessly, so this means that they do so in consideration for their own satisfaction, and are by nature selfish. So, is it moral to be greedy, or greedy in specified situations, or is there something else?
(Just posting like this to facilitate discussion. No hard feelings. :c)