It's true that John referred to himself as "the disciple whom Jesus loved" on several occasions. But here are some important facts:
First of all, John is the only one who uses this title for himself, so, technically speaking, he wasn't "known as the beloved" — it's simply a kind of nickname he applied to himself. But the next fact is that John never explains why he uses that title nor does he explain what it means.
I think we can be absolutely certain that John never intended to convey that Jesus loved him more than anyone else. I mean, this is the same man who recorded the words, "For God so loved the world that he gave his only begotten Son..." John understood the love of God and I don't believe he saw himself as a unique or special recipient of the love of Jesus.
The other thing we need to understand is that there were many sayings and statements used in antiquity that translate very poorly into modern English. I believe this is one of them. I think John was actually using this statement in place of his own name as a way of simply referring to himself as one who was loved by Jesus. Not the only one, mind you, or even the one best loved...but just simply one loved by the Lord. I think it's a name that any of us could take on after coming to the realization that we are the unlikely recipients of God's unconditional and life-transforming love. John was so taken with the idea that Jesus gave Himself as a sacrifice, that he spoke of that love in a personal and tender way. "I am one who is loved by Jesus." Whatever exclusivity we hear in that statement is, I believe, more of an irregularity of the transition from ancient to Greek to modern English.