That's very interesting - your other breeds just hang with the dog, or run away from him/her? I don't have any experience except with these Buckeyes.
I wish I could really tell the four pullets apart now as I used to. Maybe I will band them? Butters, with the lightest coloring, used to be a lead chick, and I think it is she who has the courage to face off with the Fur Chicken (cat) Big Boy on a regular basis. Big Boy is a drop-off stray who was/is basically an indoors cat, really has no clue how to hunt, and knew the chicks since Day One. He would sit and watch them. As they grew I let them come close while holding him, and let him move away as he wanted. Now they are taller than him and fiercer than him. So he backs away, poor thing, all he wants to do is hang with me while I'm with them. Sometimes when I go over to pet and reassure him, that just emboldens Butters to face off again and drive him further off unless I can pick him up first. Even then he asks to be put down so he can run away! Big Boy gets his cuddles and lap time in the Big Coop.
When Butters sees whom I think is Popcorn - she was a chick with indominable spunk - or it could be Peanut, who used to hang with Butters - coming over to join her (they used to hang together, maybe still do? There are two foraging pairs) they both setout at a full run to ambush poor Big Boy. He is relegated to about 30 feet away. Then they leave him alone. But he still wants to sit and watch them. Sometimes they run at him from behind too - and he hears their feet approaching only when very close. He crouches for some kind of unknown assault while turning to look, and usually makes a quick bounding getaway at the last minute.
It does sometimes look like play, does your dog consider it play? The Buckeyes have a great curiosity about this cat too. There have been a few times Big Boy appears to feel safe enough among them as long as he is sitting on something just above their height. They tend to tolerate him like that. However, that can leave his gently moving tail quite vulnerable. Man, that thing hanging down with the end curling a bit, moving this way and that....what a tempting thing for a chicken to investigate! Twice now, I've seen two or three of them approach it slowly and warily, quietly stepping left and right, but ever closer. They can't help themselves it seems, it is so very interesting. Each time I observed this, Big Boy would suddenly notice these Ninja Chickens right behind him and leap away to safety. Good thing, all Fur Chickens must protect their tails!