If it is owned and operated by the local community and their duly elected representatives. If you can call the people who run your school to talk about your school, and it's not a long distance call, that might be a public school. If your school is run by a board of directors who must all stand for election by the taxpayers who foot the bill for your school, you are probably a public school.
If it is operated with financial transparency. If any taxpayer can walk into the main district office and request a copy of the budget and receive a copy, that's a public school system. If you have the opportunity to call or meet with those local elected board members t argue about how your tax dollars are being spent, it's probably a public school.
If it cannot turn down a single student from your community. Your school system may sort students into specialized schools, or it may pay the cost of sending Very Special Need students to Highly Specialized schools, but it cannot ever deny unilaterally responsibility for students just because they cost a lot of money or require specialized programs or just fail to behave compliantly. If your school system can't wave a student off and say, "She's not our problem," your system is probably a public school system.
If it provides students and staff the full amount of appropriate legal protections, it could be a public school.
If it operates in a building owned by the taxpayers, it could well be a public school.
If it operates under the assumption that it will stay in operation for as long as the community wants it there, and plans to be there for generations irregardless of how well the "business" is doing, it is probably a public school.
And if your school does not make budgeting choices based on the notion that the less money spent on the students, the more money some private individual gets to pocket, that's a healthy sign of a public school.
If it meets all these standards, then your charter school is indeed a public school. If not-- well, it may be a lovely, delightful, popular school, but it is not a public school. A private school that collects public tax dollars is still a private school.
And if your public school system no longer meets these standards (if, for instance, your elected local board has been replaced with state or mayoral control, that's a sign that somebody is trying to privatize it, and may have partially succeeded.
You can say that a pig is a cow. You can dress it up in a cow suit and just keep insisting over and over that it's a cow, correcting everyone who says differently. But at the end of the day, when you butcher it, you still get pork.