These days I would say:
Underclass
Working Class
Lower Middle
Middle Middle
Upper Middle
Upper
Aristos
Royalty
I think we need the distinctions between types of MC because the middle class is such a huge and all-encompassing tranche these days, which defies specific classification otherwise. What used to be the WC is much much smaller than it once was, and we know have the underclass beneath it. There was a time about 30 or more years ago, when you could know exactly what to expect from someone who was middle class, but these days that is not the case.
People have moved from being previously WC to MC, purely by dint of a good income, especially if they earned it though trade, so they would probably be called lower MC. WC people who have been very well educated, and have probably gone on to have professions will become middle MC as adults, even if they become extremely affluent they are rarely though of as more than MMC. At least not by the UMCs, for whom it is not about income or even education, but about background, accent, values, subtle social indicators and so forth. A high income helps, obviously, but no amount of money will make you UMC if you are not.
Whereas I think the Upper Middles are born UM and remain UM, irrespective of their academic achievement or their financial trials and tribulations. Once you have been UMC even if your income drops to a very meagre level, your status as UMC rarely does.
Social mobility tends to happen from WC through LMC to MMC, but rarely outside of that.
The underclass tends to stay where it is (on the whole) as does the UMC and above, whereas there is loads of movement between WC and MMC.
I am not sure it can be defined by income alone.