http://en.wikipedia.org/wiki/State_school
In the above article, under the title "General Characteristics," the conception that Public Education means التعليم بمصروفات is corrected since the accurate designation of the latter is "Publicly Funded education" not "Public Education". Other than that, and according to the same article, Public Education is generally synonymous with State Education.
On the other hand, General Education is not a type of education as much as it is a governmental program offered in some foreign countries. It cannot be translated to mean التعليم العام as we understand it in Arabic. It appears that التعليم العام refers either to State Funded Education, i.e, Public Education, or to Basic and Secondary Formal Education, depending on context. For all this Public Education seems to be more right to me. I am not perfectly sure, and I appreciate your comments and explanations.