Both of your premises are correct, but I would not consider them real grammatical rules, because what you have in both instances is essentially an elliptical construction: "that/which is" and "After" have been omitted from the first and second sentences, respectively, for the sake of brevity.
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big dream-
1 If the I add 'a relative pronoun' to the participial phrase and it still has the same meaning, then the phrase has an adjectival phrase and vise versa.
2 If the partcipial phrase has an adverbial meaning and this can be determined by adding the suitable subordinator, then it has an adverbial function.
3 Participial phrase with adverbial meaning is