Hey, AC. I think you are right that nobody has really answered your question as you posed it, but they do THINK that they have. So let me try to couch one of their answers in your question... The fundamental difference between buying labor and buying cigarettes is that the money for the cigarettes just goes into the coffers of the company (and that of the sin tax goes to the government), but the money for the LABOR goes to a living person, who in turn PUTS THAT MONEY BACK INTO THE ECONOMY by spending it. So when minimum wage goes up, spending among consumers goes up, so demand for goods and services rises, and businesses can prosper thus hiring more people at higher wages. And just because we don't know exactly where the cutoff is between a GOOD rise in minimum wage versus a BAD one, that doesn't mean that a cutoff doesn't exist; our economic models are just too rudimentary to map it out exactly.
That's what they are arguing against you, but I say to them that this assumes too much. For instance, if you are making minimum wage--whatever it is--and you find yourself with some money to spare, if you have any sense at all you will SAVE it, so that you have a rainy day fund or use it as capital toward getting/finding a better job!!! The arguments also ignore the fact that not everybody who WANTS a job NEEDS one that pays a living wage, like high school students in the summer. Instead of learning how to participate in the economy, young, unskilled, casual workers like these are generally priced out of the market. But the dark side to the issue is that some will enter the marker NONETHELESS, thus competing with those who ACTUALLY need to make enough money to live on. Then there is black market labor. I had a friend in high school who worked construction, hauling bricks, among illegal immigrants and people on welfare. Illegal labor has no minimum wage. Raising the minimum wage increases the black market for labor, thus giving companies with high demands for unskilled labor further incentive either to hire people illegally, move to a country where it costs less to hire people, or go out of business.
tl;dr: having a minimum wage means having a black labor market, and raising the minimum wage hurts those it is meant to help.