Learn English – A friend of me vs a friend of mine

I think you're largely on the right track.

However, I would assume that “A friend…” was indeed a friend of yours – not a friend of a friend. However, the use of a (instead of the more familiar and intimate my) might suggest the person is more of an acquaintance than a close friend, but that's not necessarily a given.

I don't think “One of my friends…” is too wordy. It might be especially useful if you remember hearing something from someone, but you can't quite remember which friend gave you the information.

Also, instead of “told me”, I might be more inclined to simply use “said”.

Lastly, I think there's one viable option you've left of your list:

A friend of mine said...

Possessives that go after the noun they modify with an "of": (mine, yours, ours, theirs, his, hers, ...) can also be used with linking verbs:

It was mine. ==> It was a book of mine.

It will be yours. ==> It will be a book of yours.

Possessives that go just before the noun they modify (my, your, our, their, his, her, ...) aren't used in the "of" form, except for special cases where adding an extra 's' at the end of the possessive doesn't make sense because it already has an 's' (for example, his), so the same word is used for both roles: (his book ==> a book of his).

Another collection of possessives (mine, yours, ours, theirs, his, hers, ...), which go after the noun they are associated with and using an "of", cannot be used just before the noun unless they are a special dual-role possessive (his). These can also be used with linking verbs, as in "It was mine." ... or ... "It will be yours."