If you want an action to start if a button is pressed, and continue until another button is pressed, you need to separate the button press reading from the action.
When the button is pressed, set a flag indicating which action is to occur.
After reading all the buttons, perform the correct action.
int action = 0;
void loop()
{
if(digitalRead(buttonPin1) == HIGH)
action = 1;
if(digitalRead(buttonPin2) == HIGH)
action = 2;
switch(action)
{
case 1:
animation1();
break;
case 2:
animation2();
break;
default:
break;
}
}