Hi, As a generalization, I would say that the word is most often used in the sense of 'a personal admission'. However, I would normally look at the context to clarify the meaning. For example, in the text from which you have extracted this sentence, is the writer "present' in a first-person kind of way?
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I wonder if there are any criteria to distinguish between the two.I doubt it. It's more a matter of common sense. In your example, I would say that admittedly refers to a generally accepted idea. These details themselves are valueless (as everyone agrees); their value ...