Malcolm,
1. So payment is irrelevant1, it is compulsion that matters. Good. If someone volunteers to work that is not slavery. In this case I'd say the students are compensated with an education2.
2. You misunderstood the example. I proposed to assume compulsion3 and was drawing attention to your linguistic games4. You choose to highlight that the words homework and schoolwork contain a common root. So does guesswork and housework5.
You've advocated for home schooling6, yet I believe it meets your definition of slavery as well7. If your child chooses not to participate in home schooling but you force her then she is apparently a slave.8
1. If it were irrelevant I would not have said compensation is a matter of degree. Compensation is also an indirect measure of compulsion, as I indicated with the comment "The test of whether or not compenstion is enough to make labor adequately paid or not is whether or not the worker stays on the job without duress (compulsion)".
Slavery is a matter of degree because both compulsion and compensation are matters of degree. As Randfan indicated, this is not really a dispute over definitions but an empirical question about the effect of various institutional structure. Laws and institutions shape incentives.
2. The prevalence of this assertion reflects survivor bias and narcissism. Survivor bias because people who did well in school are overrepresented among the ranks of policy makers and people who participate in internet forums. Narcissism because people generally want to think well of themselves and so elevate their particular sort of success. Many people do not regard what schools supply as "compensation".
3. But you introduced a confused concept of "compulsion": (Niche): "
If guesswork is involved in completing a puzzle does that make it a potential candidate for slavery as well, assuming I can show that it was compulsory as I had no really logical way to solve it otherwise". Who compels you to accelerate toward Earth with a force proportional to the product of your mass and the Earth's mass, divided by the square of the distance between their centers of mass? This is the politics forum. We discuss the contribution that humans make to society, not gravitatioin, the solar flux, etc.
4. Can we do this without insults? If not, then I suggest that you're playing linguistic games with the above confused conception of "compulsion"?
5. Certainly housework is work. The term for indoor slaves was "domestic" or "household slaves".
6, 7, 8. I'm an advocate for a reduction in the use of remotely-directed interpersonal violence and threats of interpersonal violence (i.e., government) in the education industry. I'm mostly indifferent as to what arrangements evolve as the State retreats from the education industry.
We have already noted the anomalous position that children occupy in economic theory: they are halfway between "property" and "rational economic actor" (the two classes of things in the stripped-down model universe of economic discussion). For each child, somebody or some body will decide how that child spends the time between birth and age 18. Parent control of education does for children what the system of title and contract law does for other resources: unite local knowledge, and control with the incentive to use resources in socially beneficial ways.
