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In the context such as "free press", it means libre from censorship, "gluten-free" means libre from gluten and so on. Then there is "free stuff", why is the same word used? I don't think there's any difference in meaning, although "free of charges" is much less common than "free of charge".

Regarding your second question about context: given that English normally likes to adopt the shortest phrasing possible, the longer form "free of charge" can be used as a means of drawing attention to the lack of demand for ...