I don't see why it matters what you call yourself. I just say I'm not working. In real life, I have never heard anyone describe themselves as a SAHM, though most if the women I know don't work. This kind of discussion only exists on MN.
In short, if you make the decision to become a SAHM it's because a) you want to and b) you can. That's all there is to it.
It is a lifestyle decision to focus on your family without the interference of a job. So, in that respect, you are a SAHM whether your DC are at school or not because that is the lifestyle you have chosen. If you don't have a job, you are likely to be doing stuff in the interests of the family during school hours anyway - you just do more if it than you could otherwise.
For instance, if one of mine ring me in a panic because they have left their sports kits or whatever at home, I can take it in. If I was at work, they would have to suck it up. If they need tights, or costumes, or something for a school project, I have time to think about it and get the stuff in. I have 4 DC in 3 different schools. If one is sick, they tend to get it one after the next, but no worries about time off. Similarly differences in term dates are no problem. I have more energy for them after school because I can clear my head when they are at school. The evenings with 4 DC are hectic with homework, clubs and all the "issues" and I rarely sit down between 3.30 and 10.30pm. When the DC were babies and in bed by 7pm, evenings were more flexible or free time. Now that flexible / free time has shifted to school hours. Life is still busy and no "SAHM" I know is ever bored, whether the DC are school-aged or not.
OP - If your friend is lazy and has a messy house, she would be the same in any situation because that is her personality.