The background is Aristotle's distinction between predication and inherence.
The predication-inherence distinction may ... understood as Aristotle's first attempt to distinguish what is essential from what is incidental to the nature of an individual thing. What is predicated of a subject is essential to its being what it is and what is present in a subject is incidental to this. Aristotle does not exactly say that he is trying to distinguish what is essential from what is incidental to the nature of an individual thing, but his examples and statements strongly suggest that the attempt is being made. The
difference between an affirmative statement of predication and an
affirmative statement of inherence ... is basically
that in the first a particular is mentioned and the nature or part of
the nature of that particular is specified, while . In "Socrates is (a) man" the nature
of Socrates is specified, but in "Socrates is white" the nature of a
particular present in Socrates is specified, namely, what we call his
whiteness. (James Duerlinger, 'Predication and Inherence in Aristotle's "Categories"', Phronesis, Vol. 15, No. 2 (1970), pp. 179-203 : 181.
Some of the language here is unhelpful : 'in the second a particular
is mentioned and the nature or part of the nature of another particular
which is present in it is specified'. But the basic picture is clear :
When 'is a man' is predicated of Socrates, his essential nature is specified; it is the essential, not incidental, to nature of Socrates that he is a man. In other words, Socrates would not be what he essentially is if he were not a man. In contrast when 'is white' is used to describe Socrates, whiteness is merely incidentally 'present in' Socrates. He could be pink or blue and still be Socrates.
To take up your comments : whiteness is 'present in' Socrates but not as an essential part of his nature - in this sense 'not as a part'. Man or manhood is predicated of Socrates but whiteness inheres in men and other things. Not in all men but only in some, incidentally including Socrates. Whiteness 'cannot exist separately from what it is in' : no men and other things, no whiteness.
James Duerlinger, 'Predication and Inherence in Aristotle's "Categories"', Phronesis, Vol. 15, No. 2 (1970), pp. 179-203.
J. M. E. Moravcsik, 'Aristotle on Predication', The Philosophical Review, Vol. 76, No. 1 (Jan., 1967), pp. 80-96.