Any increase in demand from a higher minimum wage will represent a decrease in demand somewhere else. There is no change, and you're displaying your ignorance of economic subjects by continuing to assert otherwise.
A dollar is a dollar. Neither "helps" the economy by producing new demand. It's a phony argument.
That argument refers to State shaping of economic behaviour during crisis, such as a depression. Redistribution is always a good thing for the working class, but it is generally good for the economy during bad times as the poor will spend their money, while the rich will not - they will keep it under the mattress.