I did a very brief dig into historical justifications for the minimum wage. One thing I wanted to check is if "living wage" was a more modern term, but both were in use 100 years ago.

The writing of the time speaks of minimum wage as a worker health and safety issue. The first minimum wage laws in the U.S. only applied to women and children. As part of the labor movement, it was often described as a type of collective bargaining. Where it was seen as providing a living wage was not directly at the minimum wage but because of the inflation effect it would have on more skilled wages. That by raising the wage floor, the industries in which a head of household would be employed in would have to raise their pay accordingly to that of a living wage.

So there's truth that minimum wages weren't introduced to be living wages themselves. At the same time that was 100 years ago and we shouldn't be writing laws for today based on the assumptions of the 1900's.