I haven't actually done a REU so not too sure about that one, but I know a friend who did one and it was a combination of classes and research. I did the NIH summer program for two summers a few years ago and it is definitely a real research experience, but forgot it's not open to international students. If students are expected to present a poster at the end of the experience, I'd say it's legit (and depending on whether you find anything substantial in the poster, that will go towards a publication).
Personally, I've been taking "independent study" courses the past 3 semesters that allow me to take 1 less class each semester and use that time towards research (in addition to full-time summers with the same research group). I think building strong connections with research advisor and getting good rec letters from them is a really nice boost to application. When I applied, I had rec letters from research advisor for epidemiology research (worked with ~9 months), advisor for genomics research (worked with ~2 years), and a statistics professor (took 2 classes with). So far, I'm having more success with acceptances at biostatistics programs and statistics programs that aren't as theoretically focused, so perhaps research experiences not in pure stat are less useful if applying to a more traditional statistics school.