Quite so. A more modern translation (English Standard Version) goes like this:
On the contrary, when they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised (for he who worked through Peter for his apostolic ministry to the circumcised worked also through me for mine to the Gentiles), and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles and they to the circumcised. Only, they asked us to remember the poor, the very thing I was eager to do.
And, moreover, even in the King James version, the true meaning of the one sentence (readily available by context from the the verses surrounding the cherry-picked one) make clear that the meaning is as I had suggested: Peter would work among the Jews, Paul among the Gentiles.
Do you think this means that they were impressed with his style, or demeanor?
"And I went up BY REVELATION, and COMMUNICATED UNTO THEM THAT GOSPEL which I preach among the Gentiles, but privately to them which were of reputation, lest by any means I should run, or had run, in vain". (Gal. 2:2).
"Now to him that is of power to stablish you ACCORDING TO MY GOSPEL, and the preaching of Jesus Christ, according to the REVELATION OF THE MYSTERY, which was KEPT SECRET SINCE THE WORLD BEGAN, But NOW is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known to all nations for the obedience of faith:" (Rom. 16:25,26).
The Gospel of the Uncircumcision -GRACE
Paul's gospel deals with things never prophesied but kept a secret and concerns a body of believers- called the Church the Body of Christ.
We do not have to guess what the gospel of Paul was, he tells us:
"Moreover brethren I declare unto you THE GOSPEL WHICH I PREACH UNTO YOU..."(1 Cor. 15:1). what was it?
"For I delivered unto you FIRST of all that which I also RECEIVED, how that Christ died for our sins according to the Scriptures, and that He was buried, and that He rose again the third day according to the Sriptures." (1 Cor. 15:3,4).
"Ifye have heard of the Dispensation of the GRACE of God which is GIVEN ME to you-ward: how that BY REVELATION He made known unto me, the mystery"..(Eph. 3;2,3,5,6,9).
The gospel of the uncircumcision committed unto Paul teaches completeness in Christ. You cannot add anything to that which is complete
"For in Him dwelleth all the fullness of the Godhead bodily.And YE ARE COMPLETE IN HIM, which is the head of all principality and power: IN whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcisin of Christ; Buried with Him in baptism wherein also ye are risen with Him through the faith of the operation of God, who hath raised Him from the dead. And you, being dead in your sins and the uncircumcision of your flesh, hath He quickened together with Him, HAVING FORGIVEN YOU ALL TRESPASSES." (Col. 2:9-13).
Does this Gospel of the Grace of God, given to Paul, sound like the Gospel of the Kingdom, given to Peter and the 11?
The Book of Acts is a transition period. From Law to Grace. From a Kingdom of believers to a Body of believers.