Hmm, I suppose if you use "my friend", it is better to use it when you're speaking to a person that is NOT your friend. The reason for this is because it suggests that the person you're talking to is NOT your friend. However, if you say something like "I went shopping with my friend from Chicago" and the person you're talking to is NOT your friend from Chicago, then it would not be rude. I hope that makes sense for you, I can clarify it if you want.
So in short, use "my friend" only when speaking to a person who is not your friend (such as family members, acquaintances, etc)
You can use "a friend" or "a friend of mine" anytime you want.
Remember, this is just how I would see it. If you're brought up around a different environment you have different tendencies when speaking English.