It seems altogether probable that the apostle John refers to himself as "the disciples whom Jesus loved" and "another disciple" for two reasons. In the first place, the fourth Gospel is focused on love as the motivation of Christ in the work of redemption. In other words, John is insistent on centering on the theme of the love of Christ for His people. This is why he says such things as: "Now Jesus loved Mary and Martha and Lazarus," and "Having loved His own who were in the world He loved them to the end." But there is, it seems to me, another reason for the intentional anonymity with which John refers to himself. The saying of John the Baptizer, "He must increase, but I must decrease," (John 3:30) is only recorded in this Gospel. The apostle John finds in these words something of the goal of all Christians (i.e. to be lost in the wonder, praise and glory of Jesus Christ to such an extent that we would forget about ourselves and our own reputations). This, it seems to me, is the reason why he speaks of himself by the simple designation, "another disciple," at the climax of Christ's sufferings (see John 18:15-16), rather than actually drawing attention to himself. In this manner, John wants to direct our attention to the One to whom all glory is to be given--namely, the Lamb of God who takes away the sin of the world. He is, in the most intentional way, saying with John the Baptizer, "He must increase, and I must decrease." Here is another related post.