When did the church begin to call itself "Catholic"?

Ignatius of Antioch d. c. 110 ADLetter to the Smyrnaeans 8.2

Wherever the bishop appears, there let the people be, just as wherever Jesus Christ is, there is the catholic Church.

The first use of "catholic" for the Church, about 110. Here it means whole or universal.

Martyrdom of PolycarpMartyrdom of Polycarp Preface; 8.1

The church of God sojourning at Smyrna to the church of God sojourning at Philomelium, and to all the communities of the holy and catholic Church in every place ... remembering ... the whole catholic Church throughout the world.

In 16.2 Polycarp is called "bishop of the catholic Church in Smyrna": here the word already marks the true Church against sects.

Cyril of Jerusalem c. 313–386 ADCatechetical Lectures 18.23

It is called catholic because it is spread through the whole world, from one end of the earth to the other; and because it teaches universally and completely all the doctrines that people should know, about things seen and unseen, in heaven and on earth; and because it brings every kind of people to godliness, rulers and ruled, learned and unlearned; and because it treats and heals universally every kind of sin done by soul and body.

Cyril explains the creed's "one holy catholic Church" to candidates for baptism.

Augustine of Hippo 354–430 ADAgainst the Letter of Mani called Fundamental 4.5

Finally the very name Catholic holds me, which not without reason, among so many heresies, this Church alone has kept; so that, though all heretics want to be called catholics, yet if a stranger asks where the Catholics meet, none of the heretics dares to point to his own basilica or house.

Just before, Augustine lists the agreement of peoples, the succession from Peter's see, and miracles.

Vincent of LérinsCommonitorium 2.6

In the Catholic Church itself, the greatest care must be taken that we hold what has been believed everywhere, always, and by all. ... what the very force of the name and its meaning declares, taking in, as it does, nearly all things together.

Vincent, writing in 434, gives the classic test of the name itself: a belief is catholic when it is the belief of the whole Church in every place, from the beginning, and by the consent of all, because the word catholic itself means taking in nearly all things together. A local or new teaching that fails the test is not catholic, however loudly it claims the name.