For .edu I got my answer from the Forum. Only yesterday there was a similar thread.
'..To obtain an Internet name in the .edu domain, your institution must be a postsecondary institution that is institutionally accredited, i.e., the entire institution and not just particular programs, by an agency on the U.S. Department of Educationís list of Nationally Recognized Accrediting Agencies..'
Sometime ago, kirameki.edu was actually a website for a computer game called "Tokimeki Memorial" (Kirameki High School was a location featured prominently in that game), rather than being a website for an educational institution as one might expect. I don't know how they managed to arrange that.