In my reading of the Gospels, I have not noticed any derogatory or sectarian statements using the word "Israel", such as in:
Matthew 10:6 ... but go rather to the lost sheep of the house of Israel.
Matthew 15:24 But he answered and said, I was not sent but unto the lost sheep of the house of Israel.
Matthew 19:28 And Jesus said unto them, Verily I say unto you, that ye who have followed me, in the regeneration when the Son of man shall sit on the throne of his glory, ye also shall sit upon twelve thrones, judging the twelve tribes of Israel
However, there are many derogatory or sectarian statements using the word "Jews", such as:
John 6:52 The Jews therefore strove one with another, saying, How can this man give us his flesh to eat?
John 5:16 And for this cause the Jews persecuted Jesus
John 7:1 ... the Jews sought to kill him.
John 7:13 Yet no man spake openly of him for fear of the Jews.
John 8:44 Ye are of your father the devil, and the lusts of your father it is your will to do
In John 7:41-43, a dispute arises due to Jesus being from Galilee rather than from Bethlehem even though the Gospels of Matthew & Luke introduce (questionable) birth stories tracing the ancestry of Jesus back to David & his birth to Bethlehem, despite no words of Jesus I could find mentioning this. I note 1 Timothy 1:4 states:
...neither to give heed to fables and endless genealogies, which minister questionings, rather than a dispensation of God which is in faith
My questions fall into the following general categories:
(1) Do the words "Israel" and "Jews" have a different meaning in the Gospels?
(2) Are these "Israel" versus "Jews" statements of Jesus based in a sectarian dispute between religious remnants of the former Kingdom of Israel (of which Jesus of Nazareth represented) and the Jews of the Kingdom of Judea?
(3) Was Jesus a "Jew" or did Jesus represent another sect within the Biblical or otherwise local Judaic religious traditions (such as Pharisees, Sadducees, Essenes, etc)?