To do this, take hold of the stopper with your fingers, and try and turn it. If you do not feel any give, or if there are no signs of movement (other than spinning on its axis), then your stopper will require more work.
The third step requires your flashlight. Use this to locate the horizontal plunger rod that connects the strap to the stopper itself. Once you have located this, then look for the rod that connects the vertical strap.
If you have followed these steps correctly, then you should be able to release and remove the bathroom sink stopper from beneath the sink, and you should be able pull it out without any issues or resistance.