Tracy Jennings, Works at Noodle and here to help!
The wide range of school choice options can be very confusing - check out Noodle's guide to school choice in the South for more information - but to answer your question succinctly, Easley Elementary is both a public school and a magnet school.
(Like charter schools, all magnet schools are public. They are different because these types of schools don't follow district lines.)
For Easley, no students are selected not by how close they live to the school. There is a lottery system, but certain factors influence the likelihood students gain entry via this system, such as having a sibling who also attends the school. You can read more on the Durham School District website.