And this was a special event... it happened in that order to convince the doubting Jewish believers that Gentiles were accepted by God....
Where does the Bible teach that that was the reason, and the exclusive reason. In Acts 8, there is some gap of time between the Samaritans believing Philip's preaching and being baptized and their receiving the Holy Spirit/the Spirit falling on them. In Acts 19, there was a gap of time between the approximately 12 disciples in Ephesus being baptized and the Spirit coming on them... however long it took between baptism and Paul laying hands on them.
If all scripture is profitable for doctrine, and we only have so many scriptures on the topic, are we to dismiss the lessons of certain scriptures as just being a special case, and saying 'This was the one and only reason that happened'? If the Bible doesn't teach that X was the only reason Y happened, why should that be part of our doctrine?
The way I've heard it explained is that "apostles" were actual witnesses of Jesus.... meaning, they SAW him and associated with him. Paul was made an apostle because of his meeting Jesus on the road.... he was an eyewitness to Jesus..... that gave him the status of apostle.
Saying that George down the road is an "apostle" because he received that "gift" is not correct. George has not seen Jesus, or associated with him. All of that stopped when the last of the original apostles died out. At least that is what I understand scripture to say...
According to Peter, whoever was to replace Judas as the 12th apostle not only had to have seen Jesus, but also had to be going in and out with the apostles from the time of John. Paul did not fulfill that requirement, but he was an apostle. Acts calls both Paul and Barnabas apostles in Acts 14:4 and 14. Paul seems to allude to Barnabas' apostleship in I Corinthians 9, also. We don't know whether Barnabas was a witness of the resurrection. But in Acts 13, we read that the Spirit spoke and said to separate Barnabas and Saul for the work to which he had called them. Then they left, being sent out by the Spirit.
Paul, Silvanus, and Timothy wrote II Thessalonians according to 1:1. In chapter 2:6-7, they refer to themselves as 'apostles of Christ.' Now it is possible that Silas/Silvanus had seen the risen Lord prior to the ascension, but scripture doesn't tell us that much about his background. We know that he was a prophet that the apostles and elders had sent to deliver a letter with instructions for Gentile believers. But Timothy? It is extremely unlikely that he saw the risen Lord. I Corinthians 3-4 seem to be referring to Paul and Apollos as apostles. Those two are the subject when Paul wrote about God setting forth the apostles as a spectacle to the world, and to angels and to men.
Paul calls Epaphraditus 'your apostle'. That might mean he was the messenger from Philippi to Paul, or that he was 'their apostle' in some other sense. He was a co-laborer and fellow soldier with Paul. The church in Corinth had 'apostles' who would deliver money also.
Historically, 'apostle' was also used of certain church leaders, usually bishops, who were influential in spreading the gospel to new territories and people-groups: Gregory apostle of Armenia, Oscar apostle of the North, Patrick apostle of Ireland, the twelve apostles of Ireland, Cyril and Methodius apostles to the Slavs.
The definition you shared shows up on Protestant apologetic defenses of the Bible and descriptions of apostles. I don't see that as the definition of apostle as derived from studying the texts of scripture.
Btw, some people think Paul was the replacement for Judas as one of the twelve. But that does not fit Peter's description, and Paul says that Jesus appeared to ___the twelve__ before appearing to him. Judas was dead. If Mathias and Justus Barsabas were going around with the twelve from the time of John the Baptist, they could conceivably been present when Jesus said to His disciples that they would sit on twelve thrones, judging the twelve tribes of Israel. But Paul was an apostle to the Gentiles.
Also notice that Paul wrote that Christ appeared to 'all the apostles' after appearing to the twelve, and before appearing to him? Was he including the 70? Jesus sent them out? Was he including James?
I Corinthians 15
5 And that he was seen of Cephas, then of the twelve:
6 After that, he was seen of above five hundred brethren at once; of whom the greater part remain unto this present, but some are fallen asleep.
7 After that, he was seen of James; then of all the apostles.