Again, these words are all about perspective... I think the middle class is generally defined very broadly to include most people in the country - at least when you are grouping lower middle class, middle class and upper middle class all together. "Poor" and "Rich" include everyone else. Poor can be broken into "truly poor" - which includes the homeless and people living on government programs - and the "working poor" which includes laborers, minimum wage folks and others that are just making enough to live. Above that you start with lower middle class.
To get above upper middle class to "rich" I think you have to get into the top 3-4% - folks with large investments, possibly multiple houses, no financial concerns, etc. Being "rich" is not all about income - it is about all the other stuff - assets, lack of debt, etc. I disagree with PhDeac here, I think 'rich' and 'wealthy' are the same thing.
Like I said before, a large income may allow one to live like they are rich, but, if they lose their job and are immediately in trouble, they weren't really 'rich'.