Legally, my son is a public school student, as he is enrolled in a homeschooling charter school in CA. However, I still consider us homeschoolers for several reasons. First, I am primarily responsible for his education (the school makes this very clear when you enroll). Second, I choose his course of study and curricula (we are secular homeschoolers, but could use religious materials on our dime if we wanted). Third, I choose the pace at which he studies (he is currently accelerated by several grade levels and it is not an issue at all) and do all of the teaching in his core subjects. We turn in one work sample from each subject every other month and keep a PE log. Easy peasy. Some annual testing is also optional (though strongly encouraged), beginning in 3rd or 4th grade, as I recall. We are not test-averse, so this is not a big deal for us, but I know plenty others who opt out.
On the plus side, my son is able to take optional classes (fun stuff like Lego, music, art, robotics, foreign language, and art), as well as go on field trips all over Southern California, provided by the charter. We also receive a yearly stipend for extracurriculars and books/materials. These extras have made a world of difference for my extroverted son who needs a great deal of interaction with others. Financially, it also takes some of the pressure off our budget.
Once my son reaches high school, the freedom and flexibility of privately homeschooling will likely outweigh the benefits of the charter, but I can see us staying with a charter through most of the K-8 years.