Hi Diana,
A great resource for questions like these is the Subject Definitions document (linked to here:
www.census.gov/.../code-lists.html), though in this case, it doesn't provide direct answers.
On page 123, it's implied that homeschooling is classified as private school, though weirdly it isn't stated directly: "Field interviewers were instructed to classify individuals who were home schooled as enrolled in private school."
It doesn't mention charter schools, though public schools are defined as "any school or college controlled and supported primarily by a local, county, state, or federal government." Since charter schools are supported (if not controlled) by government, I'd imagine they're considered public, but I agree that more clarity would help.
-Bernie