Working for wages is not enough to qualify: doctor, university professor, airline pilot, solicitor... All traditionally firmly middle class professions despite working for a living.
We're talking about the UK here, I've never heard that definition in the UK and any definition of middle class that excludes doctors in the UK seems way out of line with how the term is generally used in my experience.
I have worked in a vast multitude of jobs, some manual, some temp office, some salaried, some self-employed, some company director.
Never in the UK have I ever heard of people making a distinction between a Wage and a Salary. They're the same thing.
Middle class means something completely different than the definition you're defending. Skilled office workers are firmly in the UK middle class (despite my Thatcher hating friend's assertion that he's not). It might be argued that unskilled office workers, like call centres, are now working class jobs.
the distinction, in american english, is that a wage is a per-hour thing, whereas a salary is paid for a period of employment (usually a fortnight or a month), independent of the time you actually spent working within that period.
Wages vs. salaried worker i.e. being paid weekly, possibly cash in hand for hours worked vs. a regular fixed monthly bank payment based on an annual salary.