I'm still not sure why I can't get it to do as initially thought out.
Because you thought it out wrong. Begin time is set to millis every time in the loop when button1State is LOW.
So later on when you ask
if((beginTime >= timeInterval) && (button1State == LOW))
This is always true once millis has exceeded the fixed value in timeInterval, because millis never resets back to zero, well not for about 48 days anyway, so it constantly prints out holding.
And what do you mean by keep track of the state,
At the end of the loop make a copy of button1State in another variable called say lastButton1State
why would i need to do that?
To see if their has been a change in the button state since last time, so you can detect an edge, when the button has become pressed and not just a level, is the button currently being held down.