It's official: according to the US Census Bureau, most charter schools are not "public":

Charter Schools

The data in this report include only those charter schools established and administratively controlled by another government entity (e.g., universities, cities, counties, or public school systems). The data for these “public charter schools” are collected as separate, individual units, or are included with the data for their chartering government. Charter schools that do not meet the Census Bureau criteria for classification as a government entity are considered “private charter schools” and are not included in this report.

In order for a charter school to be classified as a “public charter school,” it must meet the same requirements as any other government. It must be an organized entity, with substantial autonomy, and governmental character. Typically, if the school board is appointed by public officials, then the charter school would be classified as governmental. A few “public charter schools” are run by public universities and municipalities. However, most charter schools are run by private nonprofit organizations and are therefore classified as private. [emphasis mine]

Hey, it's not me saying that charters aren't public; it's the Census Bureau. And the National Labor Relations Board. And legal scholars. And education policy scholars.

Taking public money doesn't automatically make you a public entity; if it did, Halliburton would be its own branch of the military.

NOT a public school.