The terms 'Jew' and 'Israel' are entirely synonymous in the context as they are merely different names for the same people. Different writers used these interchangeable terms at different times as it suited them, with no particular significance in mind.
The quotes above are all relevant to the question, but the questioner should understand that the title, "King of Jews" was posted above Jesus by his persecutors, not to honor Him, but to mock Him.
It was common practice amongst the Romans to post the crime of the one crucified on his cross - to identify the reason for the punishment. Jesus Christ was guilty of no crime, although the Sanhedrin brought him before Pilate on false charges of sedition against Caesar.
When Pilate asked Jesus (Mark 15:2) "Are you the King of the Jews?" Jesus answered him "It is as you say." This was the only answer Jesus would give to any of the false charges.
Therefore, when Jesus was nailed to the cross (John 19:18-22) Pilate wrote "JESUS THE NAZARENE, THE KING OF THE JEWS." in Hebrew, Latin, and in Greek. The Chief Priests of the Jews read this, and said to Pilate "Do not write 'The King of the Jews'"...
But Pilate answered, "What I have written, I have written." Pilate's proclamation was not made to expose the truth, but it did just that. When Jesus comes again, He will be called King of Kings, and Lord of Lords.
So King he is not, for it is "God's Kingdom" until appointed by God when Jesus returns.