I feel like homeschool has many different meanings these days. One thing it is not, is a public, government school. That is what is so great about homeschool is that it can be whatever you need it to. That may sound scary to some people, but it is freeing for a lot of people. It is especially freeing for people whose kids did not do well in a government run system. Now there are different regulations in every state, so you would need to understand your state requirements, but your school can be easily individualized.
My feeling is that it doesn’t really matter what you call it as long as it works. Some people like to say they unschool or do relaxed schooling. Those terms are fine if you are trying to give people an idea of what your homeschool style looks like, but don’t think you need to fit neatly in a label. Don’t get so caught up in a label that you don’t do what actually works for your family. Your style may also be different for each child. One child may want to be more unschooled and lead his education while another might want a lot of structure. It is all homeschooling, so do what works best for each of your children and don’t worry about whether it fits within a certain structure. All that matters is that it works for you.