Its what it means, He Loved His own in the world
Jn 13:1
Now before the feast of the passover, when Jesus knew that his hour was come that he should depart out of this world unto the Fathe
r, having loved his own which were in the world, he loved them unto the end.
The irony is that the Scriptures you cite are all related, and have the same solution, whether John 6, John 10, John 13, John 17 etc.
There were those who already loved God prior to Jesus' incarnation, and were looking for the Messiah. These recognized the voice of the True Shepherd when He came, because they already knew God, already believed Moses, already believed the Scriptures, already were true sheep of God's flock. Jesus chose some as His disciples (and some as the 70, etc.).
Jesus when He came to earth taught them, prayed for them and loved them, and ministered to them, because it was through their love the world would believe that Jesus was sent into the world.
John 17:6
6 “I have manifested Your name to the men whom You have given Me out of the world. They were Yours, You gave them to Me, and they have kept Your word. (NKJV)
He manifested the Father's name to the men the Father gave Him out of the world. They belonged to the Father already, and He gave them to Jesus.
John 17:8
8 For I have given to them the words which You have given Me; and they have received them, and have known surely that I came forth from You; and they have believed that You sent Me. (NKJV)
Once they were given to Jesus He gave them the words which the Father gave to Him. They received those words, and now believed that the Father sent Jesus.
John 17:9-10
9 “I pray for them. I do not pray for the world but for those whom You have given Me, for they are Yours. 10 And all Mine are Yours, and Yours are Mine, and I am glorified in them. (NKJV)
Jesus is praying for them because they are in the world. He is going to send them into the world (see below). He is interceding for them for this task, not to abandon the world, but to send those who had been chosen to go into the world for their work.
John 17:11-12
11 Now I am no longer in the world, but these are in the world, and I come to You. Holy Father, keep through Your name those whom You have given Me, that they may be one as We are. 12 While I was with them in the world, I kept them in Your name. Those whom You gave Me I have kept; and none of them is lost except the son of perdition, that the Scripture might be fulfilled. (NKJV)
Jesus returns to the Father, but the disciples are still in the world. He asks them to keep them. None were lost during the time of Jesus incarnation, other than Judas.
John 17:15-18
15 I do not pray that You should take them out of the world, but that You should keep them from the evil one. 16 They are not of the world, just as I am not of the world. 17 Sanctify them by Your truth. Your word is truth. 18 As You sent Me into the world, I also have sent them into the world. (NKJV)
They are not of the world, Jesus is not of the world, Just as Jesus was sent by the Father into the world, He is sending them into the world.
John 17:20-21
20 “I do not pray for these alone, but also for those who will believe in Me through their word; 21 that they all may be one, as You, Father, are in Me, and I in You; that they also may be one in Us, that the world may believe that You sent Me. 22 And the glory which You gave Me I have given them, that they may be one just as We are one: 23 I in them, and You in Me; that they may be made perfect in one, and that the world may know that You have sent Me, and have loved them as You have loved Me. (NKJV)
Jesus is not abandoning the world. Others will believe through the word of the disciples who He sent into the world. And through these even more will believe that the Father sent Jesus. All will be one, that the world may believe that the Father sent Jesus.
So earlier in John 6 we see
John 6:44-45
4 No one can come to Me unless the Father who sent Me draws him; and I will raise him up at the last day. 45 It is written in the prophets, And they shall all be taught by God.’ Therefore everyone who has heard and learned from the Father comes to Me. (NKJV)
The Father drew those who had heard and learned from the father to Jesus, since Jesus had entered the world.
The will all be taught by God--the disciples were directly taught by God. And Jesus notes that they were not lost--other than the one doomed to destruction. They will be raised up on the last day. But they will also go into the world to teach what they were taught by Jesus.
John 13:1
1 Now before the Feast of the Passover, when Jesus knew that His hour had come that He should depart from this world to the Father, having loved His own who were in the world, He loved them to the end. (NKJV)
This is referring to the disciples, who were drawn by the Father to Jesus, who He loved, and now was sending into the world.
John 13:14-17
14 If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another’s feet. 15 For I have given you an example, that you should do as I have done to you. 16 Most assuredly, I say to you, a servant is not greater than his master; nor is he who is sent greater than he who sent him. 17 If you know these things, blessed are you if you do them. (NKJV)
He gives them an example of love that they are to imitate.
John 13:33-35
33 Little children, I shall be with you a little while longer. You will seek Me; and as I said to the Jews, ‘Where I am going, you cannot come,’ so now I say to you. 34 A new commandment I give to you, that you love one another; as I have loved you, that you also love one another. 35 By this all will know that you are My disciples, if you have love for one another.” (NKJV)
Jesus is leaving the world, but they are not, and if they love one another, as He loved them (the example of His washing their feet illustrated this), the world will know they are His disciples.
None of this illustrates the abandoning of the world.