I'm teaching my daughter some grammar lessons and ended up a bit confused about how to analyze participle phrases such as "removing his coat" in the following sentence:
Removing his coat, Jack rushed to the river.
When I was in grammar school, I was taught that participle phrases act exclusively as adjectives. Thus, my first instinct was to say that "removing his coat" is a participle phrase modifying Jack. But upon reflection, this seems a bit odd. "Removing his coat" seems to tell us less about a property of Jack and more about the way he rushed. Thus, it would seem that the participle phrase is adverbial. Moreover, it is possible to move this participle phrase around in the sentence (for example, to the end) without affecting the meaning, which seems like further support for treating it as adverbial.
I did some research about the issue and ended up finding a lot of seemingly contradictory information. Many reputable sources seem to support what I was taught in school. For example, one college writing program describes a participle as "a verbal that is used as an adjective" and analyzes "removing his coat" as an adjective.
Yet other sources state that participle phrases can be either adverbial or adjectival and suggest that the participle phrase in my example sentence is an adverb. (See, for example here and here.)
Here's my question: are there two schools of thought on analyzing participle phrases? Is this one of those cases where prescriptive grammarians think about an issue differently than descriptive linguists?
Or is there a nuance I'm missing?