Anger or Passionate Debate?

How do I know if I am right to strongly defend a point or if I am actually just being sinfully angry? That's not an unimportant question to think about. We know that the Bible consistently warns us that the anger of man does not produce the righteousness of God (James 1:20). But does that mean that speaking the truth will never be tense or confrontational?

Two passages help us in thinking through this question:

2 Timothy 2:22-26
"So flee youthful passions and pursue righteousness, faith, love, and peace, along with those who call on the Lord from a pure heart. Have nothing to do with foolish, ignorant controversies; you know that they breed quarrels. And the Lord's servant must not be quarrelsome but kind to everyone, able to teach, patiently enduring evil, correcting his opponents with gentleness. God may perhaps grant them repentance leading to a knowledge of the truth, and they may come to their senses and escape from the snare of the devil, after being captured by him to do his will."

Titus 1:10-14
"For there are many who are insubordinate, empty talkers and deceivers, especially those of the circumcision party. They must be silenced, since they are upsetting whole families by teaching for shameful gain what they ought not to teach. One of the Cretans, a prophet of their own, said, “Cretans are always liars, evil beasts, lazy gluttons.” This testimony is true. Therefore rebuke them sharply, that they may be sound in the faith, not devoting themselves to Jewish myths and the commands of people who turn away from the truth."

These two passage clearly show us two different ways to respond. And the difference seems to center around 1) the impact that this teaching / falsehood is having on the church, and 2) the closeness of this truth to the gospel.
Is this simply a "foolish or ignorant controversy," or is this a controversy that is directly about the gospel? If someone is just holding to a foolish position (e.g. saying something that just isn't logical or doesn't match up with reality), or if someone just doesn't know / understand what they're talking about, then the gentler approach of 2 Timothy 2 is warranted. Still correct them, but do it with gentleness and patience and not in a quarrelsome manner.
If someone is presenting a teaching that is actually a false gospel (as was happening with the circumcision party in Titus 1 and elsewhere in the NT), then the sharper rebuke is warranted. Paul set the example for doing this when he rebuked Peter for falling into the ideas of the circumcision party in Galatians 2. 

These two criteria may not answer all of our questions on this topic, but they do give us a helpful filter for thinking about how we should respond in a given situation.