In my understanding, non-self is the being without ego. No self, to me indicates no being.
Hmm. Both your statements are right. But the Tipitaka differentiates between "form" and "self". Anyway, understanding the concept of Anatta is core in the path to Nibbana.
Well, maybe we cant be rahat but at least we could try to understand this.
There is a segment in the Tipitaka called the Anatta Lakkhana Sutta. Try and read that. Form, mental formation, Perception, feelings, are all not self. When you believe you have a self is when affliction happens. Thus, focusing on not-self is the path to release from suffering and stress. That you can read in the Anudhamma Sutta. Anu Dhamma means "according to the Dhamma". It is evident that believing in having a self is the foundation for all kinds of Duka or suffering/sadness/stress like ego.
This understanding will affect your behaviour and Karma creates more Karma. If you read the Sulekha Sutta you see how to achieve effacement, and you will see "others will behave this way, but we will not" in the path to effacement. Then you get the Dammapada which shows how to become a so called "brahmana", and all of these hover around the understanding of Anatta.
No self, non-self, not self, are all meant by Anatta. Only, you should separate Form, mental formation, perception, feelings etc etc from this "being" if that's what you wish to call it.