NOT A TEACHER
(1) Some advanced learners, such as you, know more grammar than many
native speakers, such as I.
(2) Until I read your post, I had never heard of the word "determinative."
(3) I have patiently waited two days for someone to answer, but -- alas -- no one
has, so may I start?
(4) Professor Quirk in his super famous grammar book A Comprehensive Grammar
of the English Language says on page 326 of the 1985 edition:
Most commonly the genitive fuctions as a determinative.
(5) On page 327, he says:
There are occasional examples where the genitive acts as a modifier.
(a) The professor is referring to "descriptive genitives" (that classify
a ship's doctor.
a summer's day.
a doctor's degree.
(b) He gives this interesting contrast on pages 1335- 1336:
I visited his old fisherman's cottage. = modifier.
I visited his old friend's cottage. = determinative.
(6) You can decide for yourself whether you think that your examples fit the
definition of "descriptive genitives." If they do, then they are modifiers. If they do
not, then they are determinatives.
(7) Of course, I do not have the confidence to express my opinion.
(8) Let's hope that some knowledgeable person (teacher or advanced learner)