The question here is: is "my" a definite personal pronoun, or just a personal pronoun? I believe the answer is that in most dialects of English, "my" is a definite personal pronoun, so you should use "my friend" whenever you would use "the friend", and "a friend of mine" or "one of my friends" when you would say a friend.
Evidence for this is this Ngram, which shows that "a friend of mine" is roughly an eighth as common as "my friend", but "the friend of mine" has vanishingly small frequency. People wouldn't say "a friend of mine" if "my friend" had the same meaning (this is why people don't say "the friend of mine").
The word "my" is a definite personal pronoun for most other nouns in English as well. If you're asking about suitable attire for some event, you'd ask "should I wear a suit?" and not "should I wear my suit?" unless you only had one suit. There may be some exceptions for relatives (I think you can say "his brother", "his grandmother", even though he has more than one, but you'd probably say "a cousin of his" or "one of his cousins"). There are definitely exceptions for some body parts (e.g., it's "take my hand", not "take one of my hands", even though it's "I'll give you a hand" and not "I'll give you the hand").
In the Northeast of the U.S., I hear "my friend" used quite often as an indefinite personal pronoun. This was very disconcerting when I first moved here, but I'm used to it now. I think that here, "friend" is another exception that falls into the same category as brother and daughter.