The perils and pitfalls of definitions
It is the nature of definitions that they limit.
How ironic, it seems, that trying to pin down a definition of homeschooling could impinge on our freedom to homeschool. But that is, in fact, exactly what defining it in law does.
Let's examine, for a moment, the alternative. If we don't define homeschooling, then anyone and everyone can call themselves homeschoolers. It seems a little silly to invite that possibility, yet trying to prevent it by defining homeschooling in law brings certain regulation.
If a state does not allow homeschoolers to self-identify — and by self-identify, I mean that they decide for themselves that they are homeschoolers, rather than having the state determine for them whether they are or not — then the state is regulating homeschooling in some way. In other words, if the state is not simply taking a homeschooling parent's word for it that they are educating their children, then the state is requiring homeschoolers to jump through one or more regulatory hoops to demonstrate or prove that they fit the state's definition of homeschooling.
We can't have zero regulation if homeschoolers are not allowed to self-identify. It's a messy reality.