The term "living wage" better reflects the necessity of a floor for wages, despicable examples of predatory low wages can be found in Bangladesh and India. Many conservatives bristle at the term minimum wage. They make it sound like it's arbitrarily imposed by government bureaucrats to garner votes. But a living wage is hard to argue with.
Separately, once a living wage has been instituted and made into law, then standards of living go up, as they have in China, which for the past decade has ceded its coveted position as the factory of the world (at least in some sectors) to Vietnam, which has much lower wages. This brings me to my last point: sometimes minimum wage can scare off business. So whatever ends up happening, it needs to be a balancing act. I am no expert, but econ books are filled with chapters about this.
[1] https://twitter.com/arindube/status/1191319006088769536