App-designed back buttons VS navigation bar back button

The question is basically : Should app-designed back buttons do the same thing as the navigation bar back button ?

I am asking this because I consider doing specific back actions with the navigation bar back button, but if some Android don't have that navigation bar, then the users won't be able to do these specific actions.

For example : let's say the user is on activity A, he clicks a button, which opens a dialog. This dialog has multiple choices and when the user choses one of them, the dialog is dismissed and activity B appears with an app-designed back button at the top. Now, if the user presses this app-designed back button, he will be re-directed to activity A. But if he presses the navigation bar back button, then the dialog will re-open and he will see the choices again.

Is this a good design pattern ?