I looked at the 2nd picture you posted and realized it didn't have anything which connected to the cinch.
I don't think it's an Aussie breast collar.
The closing thing I could find that looked like that was this.
The second one actually looks to me like an English breastplate that is commonly seen in eventing
As for the style you'll like better, it would greatly depend on what you want to do. If you're just pleasure/trail riding and looking for something that will give you a bit of stability, then pretty much any style of western breast collar will work well for that. Personally, I like the pulling style collar, but not just for the stability while roping/dragging. They fit a wider range of horses and getting a correct fit is much easier on them than with your typical y-style breast collar.