I don't understand how your source came to the conclusion that Tom only has one friend. I would like to know more about this. This is the way I understand "a" -
A is used to refer to a non-specific or non-particular member of a group. For example, “I would like a beer.” The reference is not to any specific beer. The reference is to any beer. There are many beers, and I want a beer. I don’t care what kind it is. “I want a pen" This refers to any pen. I don't need a specific pen, any pen will do. To say that, "Tom is a friend of mine", is to say that I have more than one friend. The way out of this problem is to write, "Tom is my friend."
***** NOT A TEACHER *****
Hello, Gilnetter.
(1) Thank you for your kind note.
(2) I have reread my post, and I do not think that it contains any
contention that Tom has only one friend. In fact, I think that my post
mentions that Tom probably has more than one friend.
(3) Nevertheless, I think the most important point is that the authors of
that book write: "Some reference grammars ... erroneously state or imply"
that "Philip is one of our friends" and " Philip is a friend of ours" are
synonymous.
(4) The original poster wanted to know whether there was a difference
between such constructions. I tried to explain that there is. Other
people feel there is not.
***** Thank you *****
P. S. If I say, "Tom is one of my friends," then I have more than one friend.
If I say, "Tom is my friend," then it is possible that I have only Tom as
my sole friend. We do not know.
If I say, "Tom is a friend of mine," it is also unclear as to whether I have
other friends.