So what in the world do verses like these mean?
Romans 16:17 Now I urge you, brethren, keep your eye on those who cause dissensions and hindrances contrary to the teaching which you learned, and turn away from them.
Let us try these:
“Now I beseech you, brethren, mark them that are causing the divisions and occasions of stumbling, contrary to the doctrine which ye learned: and turn away from them” (ASV).
“I appeal to you, brothers, to watch out for those who cause divisions and create obstacles contrary to the doctrine that you have been taught; avoid them” (ESV).
“But I warn you, brothers, to keep an eye on those who cause divisions and temptations, quite out of harmony with what you have been taught, and to keep away from them” (NBV).
Put together: “keep an eye on those who cause divisions and temptations, |contrary to| what you have been taught, and to keep away from them” (NBV|ESV|NBV).
Greek translated "divisions" has a literal meaning "standing apart,"* as in acts of dissociating.
Christians had been taught not to cause acts of "standing apart." The act of dividing is prohibited; getting Christians to divide is prohibited.
Galatians 1:7 ...only there are some who are disturbing you and want to distort the gospel of Christ. 8 But even if we, or an angel from heaven, should preach to you a gospel contrary to what we have preached to you, he is to be accursed!
Are these people Christians?
No.
2 John 1:9-11 Anyone who goes too far and does not abide in the teaching of Christ, does not have God; the one who abides in the teaching, he has both the Father and the Son. 10 If anyone comes to you and does not bring this teaching, do not receive him into your house, and do not give him a greeting; 11 for the one who gives him a greeting participates in his evil deeds.
Are these people Christians? No.
Titus 1:10-11 For there are many rebellious men, empty talkers and deceivers, especially those of the circumcision, 11 who must be silenced because they are upsetting whole families, teaching things they should not teach for the sake of sordid gain.
What does this have to do with Christians dividing against Christians?
It sounds an awful lot like the Bible is saying that if people disagree we are to separate from them.
You have not shown me a single passage where Christians are told to divide against Christians over disagreement.
Romans 16:17 and other passages warn Christians against dividing against each other. Disagreement is never shown to be an approved exemption.
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*Vine, et al, Expository Dictionary, page 179 NT; in Mounce, Complete Expository Dictionary, page 1126.