Who counts as a heretic, and what makes someone one?

Ignatius of Antioch d. c. 110 ADLetter to the Trallians 6.1–2

I urge you, then, not I but the love of Jesus Christ: use only Christian food, and keep away from a foreign plant, which is heresy. These people mix Jesus Christ with themselves to gain trust, like those who give a deadly drug with honeyed wine, which the unknowing take gladly, with evil pleasure, to their death.

The heresy Ignatius has in mind is the denial of Christ's real flesh.

Irenaeus of Lyons c. 130–c. 202 ADAgainst Heresies Preface 2

For error by itself is not put on show, lest once stripped bare it be exposed; but decked out cunningly in a plausible garment, it makes itself seem, by its outward appearance, even truer than the truth itself to the inexperienced.

The preface to the whole work. The heretical teaching persuades because it is dressed to look like truth; Irenaeus says it must therefore be stripped and shown false, not merely denounced. The Greek of the preface is preserved in the PG 7 text.

Irenaeus of Lyons c. 130–c. 202 ADAgainst Heresies 3.2.2

Yet when we appeal to them with that tradition again, the one from the apostles, which is preserved in the churches through the successions of the presbyters, they oppose the tradition, claiming that they, wiser not only than the presbyters but also than the apostles, have found the unmixed truth.

For Irenaeus the mark of the heretic is a rival, self found truth set against the public apostolic tradition. He adds that anyone can inspect that tradition in every church and count the bishops the apostles appointed (3.3), and that the truth is therefore not to be sought elsewhere (3.4). The Greek of Book 3 is lost here; the passage survives in the Latin translation.

Tertullian c. 155–c. 220 ADPrescription against Heretics 6.2–4

Heresies are so called from the Greek word for choice, which a person uses either to found them or to take them up. That is why the apostle called the heretic self-condemned, for he has chosen for himself that in which he is condemned. But we may not indulge in anything of our own will, nor choose what someone else has brought in of his own will. We have the apostles of the Lord as our authorities, who did not themselves choose to bring in anything of their own will, but faithfully delivered to the nations the teaching received from Christ.

"Hairesis" means choice. The heretic chooses his own teaching instead of receiving the apostles'.

Augustine of Hippo 354–430 ADOn the Proceedings of Pelagius 6.18

It makes a difference how much, and from what source, and finally whether, when warned, a person corrects himself or, by defending it obstinately, turns into dogma what he had said lightly and not as dogma. So since every heretic is also a fool, but not every fool is at once to be called a heretic ...

Error becomes heresy when it is held obstinately after correction. Augustine says the same in City of God 18.51.

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's canon, written about 434, three years after the Council of Ephesus: a teaching is unmasked as a private choice when it fails the triple test of universality, antiquity, and consent. It became the classic rule for telling the faith the Church received from what a teacher invents.

Thomas AquinasSumma Theologiae II-II, q. 11, a. 1

because he does not choose the things truly handed down by Christ, but the things his own mind suggests to him. ... And so heresy is a species of unbelief, belonging to those who profess the faith of Christ but corrupt its dogmas.

Thomas builds the definition from the word itself: heresy (haeresis) is a choosing. The heretic means to assent to Christ, but picks what to believe by his own mind instead of what Christ handed down; unbelief of that kind, in someone who professes the faith of Christ, is heresy.