Ball Comes out when Pulling Pin

Always make them show you the rule in the rule book.  In this case the answer is right there in the definitions section as Bill posted.

And there is a whole series of decisions to the effect that once the ball is holed the hole is over.

Was the ball still rattling around in the cup all the while when you were walking up and taking out the stick?  Of course not.

Now if a little of the ball was above the level of the lip that would be a different story.  Then if you lifted the flag, and in the process expelled the ball without it ever coming to rest completely below the lip, then you have a problem.  If a little of the ball is above the lip because it is resting against the flagstick, what you ARE allowed to do is straighten the flagstick so the ball is no longer leaning on it, and if/when it drops in you are considered to have holed it with your prior stroke.

So unless you ball was not fully below the level of the lip when you removed the flagstick your playing partners are full of it.

But I will also add that it is bad form to flip the ball out of the hole with the flagstick.  You can damage the hole that way.

I've accidentally knocked the ball off the tee and replaced it and been told "that's three". That's BS, too.

If your ball is more than 50% over the edge of the hole leaning against the pin, under normal circumstances, laws of physics will hold that when you straighten the pin the ball will drop to the bottom of the hole, unless the green is muddy and the ball is adhering to it.

There is no penalty for doing what you did.

I thought if the player caused his own ball to move then he must play as if he took a swing.. Why is it BS? Unless that's not the rule, maybe someone can clear this up.

To get the ball to move upwards you have to exert some sideways pressure. The liner will resist it but the 1" of earth above the liner may not, particularly if the earth is wet or crumbly dry.

When the ball is on the tee it is not yet in play. The rule about moving the ball only applies to a ball in play.

If you swing and miss the ball on the tee, that is a counting stroke.

What about if I am taking a practice swing and I tip the ball? Or if I am just getting my arms loose before I take my swing and I tip it? Basically at what point is it determined to be put in play?

This is one of the common rules myths.  Once the ball is holed by definition (see the post under yours), the hole is finished, and the ball is out of play until you hit from the the next tee.  Since the ball is holed and your play for that hole is over, the act of removing the flagstick before removing the ball cannot cause a penalty.

From Definitions.

Holed

A ball is “ holed ” when it is at rest within the circumference of the hole and all of it is below the level of the lip of the hole .

No penalty since the hole was over.

See post #3

Sorry.  Read your first quote block and skipped the second.

But I did add that the hole was over.  ;-)

See post #5.