Standing water inside tub drain
That is just the water that's on the other side of the trap that's in place to keep sewer gas from coming up through the drain. Some traps are further away from the drains so you might not notice it but if the trap is located right under the drain, standing water will be there. When the spout is running it will disrupt the water flowing down the drain just because of the forced flow of water, that's normal. It's a great idea to install those hair catchers.