1. According to the US Bureau of Labor Statistics, about half of people making minimum wage or less are older than 25 years old.
2. A single person working a full-time job (32 hours) at minimum wage makes $12,064. The US federal definition of poverty for this individual would be anything below $12,880. So yes, minimum wage workers are literally, by definition, working for poverty wages.
For one, even with your statistics, half of folks employed are school-age. Also, poverty is measured by household. Confusing the math isn't a good way to make a quantitative argument.
But mostly your argument is a semantic trick around the word "poverty" to rationalize weighted terminology.
Again, I have strong feelings that employers should be conscious of the lives and prospects of their employees. For instance, having policies that lead to irregular patterns of night and day shifts that have to be unhealthy long-term. But we can have rational discussion about these issues.
The figure they used was for a household of size 1, so I don't think they confused the math and I think they made a perfectly strong quantitative argument.