Your first name and last name inside of your Member App account is used as a default when you create a new membership. When you change your name in account settings it will become the new default details that will be populated on the setup step when you create a new membership.
Scroll down the page until you arrive at two forms. First name and Last name.
Add a new first name &/or last name and click 'Save'.
Your changes will be updated immediately and now reflect in your account.