Abimelech means My father is King
Achish was able to say 'my father is king' thus demonstrating that his kingship was one which was passed from father to son rather than being a consequence of his own military conquests. Therefore Abimelech is a title - not a proper name. Like Pharaoh, Abimelech is a dynastic title.
We conclude, therefore, that the man who David 'changed his behaviour' before had the name 'Achish' but his title was 'Abimelech'.
Having established that Psalm 34 is speaking of the events of 1 Samuel 21:10 we have to ask another question.