however if the sole reason behind your letting them go is because they refused to give you two weeks notice imo it sounds like you are someone who actually believes that they manage people. This is a constant challenge for hiring managers especially. The moment the expression "I manage people" is something you say or have said you are on the wrong side of this equation.

Nobody manages people. People are going to do what they want to do. They are free to make choices, act in accordance to what they want. The more someone tries to manage people, the more frustrating and stressful it becomes. Managers manage resources. Time, projects, distribution, logistics, IT, etc. They rely on people in order to manage these things. If you think you actually manage people it would explain your reactionary decision to axe someone who gave you less than what you deemed acceptable notice.