A distinction that makes no difference.
In 380 AD under the Christian Emperor Theodosius I laid down the rule that only the Catholic Christians could define orthodoxy within the confines of the Roman Empire. It was under his reign that the first heretic was put to death.
- Pope Innocent III (1199) was the first to declare heresy to be high treason against God, having already called for the execution of those who persisted in their heresies after being excommunicated.
- In 1229 Pope Gregory IX declared that it is the duty of every Catholic to persecute heretics.
In the middle of the fifth century Pope Leo the Great commended the Emperor for torturing and executing heretics on behalf of the Church.
Waldo of Lyon provided the next major target. They gave their money to the poor and preached the Christian gospel. Waldo attracted the hatred of the clergy when he commissioned a translation of the bible into occitan, the language of what is now southern France. The heresies of the Waldensians were numerous. Having read the bible for themselves they denied the temporal authority of priests and objected to papal corruption. They rejected numerous accretions, including the Mass, prayers for the dead, indulgences, confessions, penances, church music, the reciting of prayers in Latin, the adoration of saints, the adoration of the sacrament, killing, and the swearing of oaths. They also allowed women to preach. They were excommunicated as heretics in 1184 at the Council of Verona, and persecuted with zeal for centuries. 150 were burned at Grenoble in a single day in 1393. Survivors fled to remote valleys in the Alps. Pope Innocent VIII organised a crusade against them in an unsuccessful attempt to extirpate them. They were still being persecuted centuries later. In Piedmont in the middle of the seventeenth century further attempts were made to extirpate them. Anyone in Villaro who declined to go to a Roman Catholic mass was liable to be crucified upside down, but there was some variation in the manner of killing in other towns. Some were maimed and left to die of starvation, some had strips of flesh cut off their bodies until they bled to death, some were stoned, some impaled alive upon stakes or hooks. Some were dragged along the ground until there flesh was scraped away. One at least was literally minced. Daniel Rambaut had his toes and fingers cut off in sections: one joint being amputated each day in an attempt to make him recant and accept the Roman faith. Some had their mouths stuffed with gun-powder which was then ignited. Paolo Garnier of Roras was castrated, then skinned alive. Children were killed in various ways before the eyes of their parents. Those few who escaped to the mountains were mostly killed by exposure, starvation or disease .
Saying the church didn't execute people is like saying the guy who hires a Hitman isn't liable for the death of his victim.