One last point here. That the word Vengeance means to repay a deed and not defend someone from harm, let's look at the context
Rom 12:17 Recompense to no man evil for evil. Provide things honest in the sight of all men.
Rom 12:18 If it be possible, as much as lieth in you, live peaceably with all men.
Rom 12:19 Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord.
AMP
Rom 12:17 Repay no one evil for evil, but take thought for what is honest and proper and noble [aiming to be above reproach] in the sight of everyone.(2)
Note verse 18 If possible, as far as it depends on you, live at peace with everyone.
IF possible. This is not an absolute command, but leaves room for situations where it is not possible.
Rom 12:19 Beloved, never avenge yourselves, but leave the way open for [God's] wrath; for it is written, Vengeance is Mine, I will repay (requite), says the Lord.(3)
Notice avenge and vengeance here is defined as repayment for deeds done. Not defend against a deed, or prevent a deed or stop someone from doing a deed to someone else. It's about repayment
Lastly, did anyone notice this is actually an Old Testament verse?
Deu 32:35 Vengeance is mine, and recompense, for the time when their foot shall slip; for the day of their calamity is at hand, and their doom comes swiftly.'
Deu 32:36 For the LORD will vindicate his people and have compassion on his servants, when he sees that their power is gone and there is none remaining, bond or free.
Yet within the context of the law NOT all killings were murder or wrong.
BTW know something funny? The greek word for coals in verse 20 is anthrax lol