They "... explained to him the way of God more accurately,"
"... if he found any who were of the Way,
whether men or women,
he might bring them bound to Jerusalem,"
The words above show that the teachings of Christ and the Kingdom had spread out from Jerusalem. And the men and women who believed and followed Jesus were described as people "of the Way".
How did the name come about? Believers know Jesus as The Way, The Truth, The Life (John 14'6) who preached, taught and lived God's Kingdom life.
He referred to the reality of the Kingdom of God and demonstrated it. And the Father sent the Holy Spirit for Christ's Called-Out Ones to do the same.
Consequently, Christ's people were like their Lord and expressed the Kingdom to the world (and to the powers of darkness!). Also, like Jesus, they desired to glorify their heavenly Father.
The Lord was their example. He took the earthly position of being the first, and foremost, citizen of the Kingdom of God.