You're right about the definition of public good, but public goods are not the only area in which it's rational for the government to intervene in the market. Both health care and education have strong positive externalities, and would be under provisioned if left purely to the private market. Particularly in the case of education where many of those who benefit from education are budget and credit constrained, the under provision will be particularly severe.
That's why it makes sense for the government to intervene.
That's why it makes sense for the government to intervene.