Assuming she doesn't have any developmental delays or disorders, I would have given her one warning and then left. In public it's one chance and you're out. Even when it's my own darned fault for not having nipped it in the bud because I brought her in hungry, tired, already irritated, or what have you. It may sound unfair to the other child but the drama that follows is more unfair to BOTH children.
Why? Because it creates more chances for the child that's already misbehaving to get into more severe trouble. In the beginning, it sounds as if your daughter was just burning off some energy and being silly hence the running around and squealing. The problem was when you tried to get her to sit down for the time out. Girl needed to burn off that energy! She was excited! Then Mommy joins in and CHASES! Whoo hoo! It's a game now! Except when Mommy catches her, it's not so much fun. So tantrum, screaming, more being forced into a time out, holding her still, more discomfort, lashing out, etc. She's a wreck. And she's in far more trouble than if she'd been removed from the situation in the first place.
This is how I've handled my typical children.
With my special needs daughter she can go from 0 to meltdown instantaneously, so at the first sign of a nearing meltdown (which is no one's fault, it just "is") we haul tushy out of wherever we are. My other girls know the signs and they're old enough to feel upset when people stare with reproachful and judging eyes, not knowing their sister is autistic. I've had to literally drag her out of stores and libraries in the middle of meltdowns.
On the flip side, I've had to remove my eldest daughter too, who is ADHD and ODD, for being fresh and rude and testing me in public. She likes to see how far she can push and sass and if I'll really discipline her in public. She always has. I always do. ;-)
We use natural consequences with creative discipline measures that don't include spanking.