I was taught that class is based on values:
-- Upper class valued family lineage/ancestry most
-- Middle class valued striving for more money, things, better quality of life. (Upper class generally already had the money, because they were descended from royalty and old money.)
-- Lower class valued living for that day, which means both that they're so poor that daily survival takes up a lot of their time and that they do things without considering long-term consequences, such as having parties when their paychecks come instead of paying their bills.
The media today talks about class as being almost completely equivalent to money, but I think education plays into it, too.
My definitions:
Upper class - has money or their ancestors did, values their lineage (royalty, aristocracy, nobility, and in the US, founding settlers and old money families), values privacy (not fame).
Middle class - values money, education, achievement. Achievement may be academic, business, military, entertainment. Recognition and fame are all good.
The thing is, IMO there is such a difference between the upper middle class and the lower middle class that I'd call them two different classes. Both work for a living, but the upper middles want the educational bona fides, the degrees and certificates. Lower middles train for jobs but don't otherwise value higher education. They value enjoying life in their community more.
By the way, find today's (or maybe yesterday's) Wall Street Journal. It has an article at the bottom of the front page on how French aristocrats have formed a sort of support group/union to help each other with money issues and finding an identity in today's world. I found the article quite amusing, but it really shows the upper class value of family/lineage still exists.