My guess, as someone who teaches this very age group research skills, is that they read in their research that people in Luxembourg wear "western attire." Then they either assumed they knew what that meant, or they Googled that phrase. In either case, cowboy clothing was the natural and very logical result. They didn't know that the term has another, different, meaning.
I think what they did was absolutely adorable, and I seriously doubt that their teacher was embarrassed.
For those questioning why the teacher didn't check a rough draft, I doubt that a detailed description of the clothing would have been part of that. Most likely, they simply referred to "western attire." The teacher would have seen a correct statement and not questioned whether they actually understood the meaning of the term.
As for the parents, by that age, most kids are capable of pulling together that kind of costume themselves from leftover Halloween costumes and their younger siblings' dress-up trunk, so the parents probably didn't realize the mistake until it was too late. Most kids I know, when they do group projects, work things out among themselves, only asking for help if the don't know something or need something. These kids obviously felt they had a handle on that part of the project, so why would they ask for help?
As a teacher, I would not have said a word during their presentation after they explained that their research indicated WW. I would, however, pull them aside after everyone was done and give them a quick explanation of what it really meant, using humor to let them know that they did nothing wrong, they just made a small error. And then, when all the kids and parents were gone, I'd have a good laugh over this cute, funny incident.