The noseband your seeing is either a Tie Down, or a Combination bit....NEITHER need to be used, it is personal preference and 90% of the people who use Tie Downs the horses do not need them and do it for looks.
I just use a simple (or blingy soon lol) headstall. I prefer browband because my mare LOVES to rub her head on our breaks between classes and when I jump off, so it stays on better during her antics.
When she can run a nice pattern at good speed, I might add a tie-down for her to lean on..but not to keep her head down. I know a few people who help their horse's run with them, and a majority of people who use them because they'll lose teeth or find themselves on the ground after they get knocked out..which is sad, because they shouldnt be running in the first place. Posted via Mobile Device
There is not a specific headstall you need for barrel racing. A headstall, is a headstall. There aren't "reining headstalls" or "barrel racing headstalls". Just a headstall.
Now myself personally, I want a browband headstall with a throatlatch, because I want that thing as secure as possible when I'm running breakneck speed through the pattern. I don't like one-eared headstalls and I don't like headstalls without a throatlatch. I feel like the horse could just rub it off. Plus, if one thing breaks, it is for sure going to fall off. But that's my personal preference.