What kind of movement or denomination in Christianity made being a Christian a personal/individual choice?

The earliest reference I can find to individuals confessing their faith in Christ and asking to be baptised (as an adult believer) thereby making a personal choice, dates back to the early 16th century when the Anabaptists rejected infant baptism and went about re-baptising adults who had previously been baptised as infants:

The early Anabaptists formulated their beliefs in the Schleitheim Confession, in 1527. Anabaptists believe that baptism is valid only when the candidate confesses his or her faith in Christ and wants to be baptized. This believer's baptism is opposed to baptism of infants, who are not able to make a conscious decision to be baptized. Anabaptists are those who are in a traditional line with the early Anabaptists of the 16th century. Other Christian groups with different roots also practice believer's baptism, such as Baptists, but these groups are not seen as Anabaptist. The Amish, Hutterites, and Mennonites are direct descendants of the early Anabaptist movement. Schwarzenau Brethren, Bruderhof, and the Apostolic Christian Church are considered later developments among the Anabaptists.

The name Anabaptist means "one who baptizes again". Their persecutors named them this, referring to the practice of baptizing persons when they converted or declared their faith in Christ, even if they had been baptized as infants. Anabaptists required that baptismal candidates be able to make a confession of faith that is freely chosen and so rejected baptism of infants. The early members of this movement did not accept the name Anabaptist, claiming that infant baptism was not part of scripture and was therefore null and void. https://en.wikipedia.org/wiki/Anabaptism

I also found a reference to Martin Luther (taken from his Small Catechism) on the subject of the sacrament of holy baptism, which quotes Mark 16:16:

Christ, our Lord, says in the last chapter of Mark: He that believeth and is baptized shall be saved; but he that believeth not shall be damned. https://en.wikisource.org/wiki/Luther%27s_Small_Catechism#IV._THE_SACRAMENT_OF_HOLY_BAPTISM

Although I could not find anything to suggest that Martin Luther denounced infant baptism and promoted the view that only believing and consenting adults should make a decision to come forward for baptism as a declaration of their faith in Christ Jesus, that does not mean to say such a quote does not exist. The emphasis on individual choice is a distinct Protestant characteristic which came out of the Protestant Reformation.

The emphasis on individual choice may not be a distinctly Protestant characteristic, but it is certainly a biblical one.

In both the New- and Old Covenants the biblical truth is that human choice is an important element in each true believer's faith walk, regardless of denomination or lack thereof.

Joshua's words ring just as true today as they did when he uttered them millennia ago:

And if it seem evil unto you to serve the Lord, choose you this day whom ye will serve; whether the gods which your fathers served that were on the other side of the flood, or the gods of the Amorites, in whose land ye dwell: but as for me and my house, we will serve the Lord (Joshua 24:15 KJV).

Not to minimize God's role in calling and effectuating that calling through the often mysterious working of His Holy Spirit, the biblical mandate is "call and response," if you will. God invites people to come to him and they in turn choose either to accept or reject the invitation. From Isaiah:

Ho, every one that thirsteth, come ye to the waters, and he that hath no money; come ye, buy, and eat; yea, come, buy wine and milk without money and without price.

Wherefore do ye spend money for that which is not bread? and your labour for that which satisfieth not? hearken diligently unto me, and eat ye that which is good, and let your soul delight itself in fatness.

Incline your ear, and come unto me: hear, and your soul shall live; and I will make an everlasting covenant with you, even the sure mercies of David (55:1-3 KJV).

Jesus reiterated Isaiah's call when he said,

Come unto me, all ye that labour and are heavy laden, and I will give you rest (Matthew 11:28 KJV).

Then said Jesus unto his disciples, If any man will come after me, let him deny himself, and take up his cross, and follow me (Matthew 16:24 KJV).

Notice in the second of Jesus's invitations above, the word will. Jesus laid the responsibility (i.e., the ability to respond) of coming after him in discipleship at least partially on the invitee. Combining the two invitations, and you have

• if any man will, which is every person's responsibility, and
• I will, which is God's assurance that His invitation is genuine, sincere, and will be honored

The decision to become a follower of Jesus Christ is a matter of choice. Regardless of denomination--Protestant or Catholic, when a baby who is baptized has reached the "age of accountability," he or she can and must choose to continue accepting God's offer to come after Him in discipleship for the "long haul." Many would-be followers of Jesus lost interest and fell away from following Him when His words offended them or when the cost of discipleship became too much for them:

Whoso eateth my flesh, and drinketh my blood, hath eternal life; and I will raise him up at the last day. For my flesh is meat indeed, and my blood is drink indeed. . . . Many therefore of his disciples, when they had heard this, said, This is an hard saying; who can hear it? . . . From that time many of his disciples went back, and walked no more with him (John 6, passim, KJV).

In conclusion, to minimize God's role in drawing people to Jesus Christ is a mistake (see John 6:35-40). By the same token, however, to minimize the importance of choosing to follow Jesus in discipleship is also a mistake. Is there an antinomy at work here? Yes, in my opinion, there is, and on this side of eternity the contradiction between principles or conclusions that seem equally necessary and reasonable (i.e. God wills and people will, God chooses and people choose) will remain a paradox.