parts of speech - In “…displaying your name” is ‘displaying’ a gerund or a participle?

I've learned so far that a Gerund can be used in 6 cases: Subject, Object, Subject complement, Object complement, Object of preposition and Object of possessive.

Participles are used as adjectives, such as; falling monkey, burning building, etc. A participle must modify something in the sentence.

In this sentence:

My phone will ring displaying your name

can someone please explain whether displaying is a gerund or a participle? I can neither put it in any of the 6 gerund categories nor can I get how it is modifying anything in the sentence.

Answer

It is only your phone that can display a name, and so it is clearly your phone that this participial phrase is modifying. However, when a participial phrase is separated from the word it modifies, it requires a comma:

My phone will ring, displaying your name.

Still, this sentence seems problematic to me because, even as it modifies the noun, it bears no relation to what the main clause is saying about the phone. For example, it makes sense to say that

My phone will light up, displaying your name

because the lighting up of a phone is what allows a phone to display a name.

Likewise,

My phone will ring, sounding a lovely Mozart melody

makes sense because the ring and the melody are clearly related.

Personally, I would phrase your sentence in this way:

My phone will ring and display your number.