What is Galatians 5:19 specifically referring to when it says dissensions & factions are an act of the flesh?

First, according to Rom 13:1-7, there is nothing fundamentally wrong with politics (However, there are definitely some bad politicians then and now.) Politics necessarily involves differences and factions.

But that is not what Paul is referring to in Gal 5 where he is discussing factionalism in the church, NOT governmental & secular politics. That is, Paul is condemning politics and faction within the church organization.

Now to the meanings of the words you highlighted. One should use a NT Greek lexicon for this rather than an English lexicon. Here are the meanings of the words from BDAG:

• διχοστασία = the state of being in factious opposition, dissension, Gal 5:20, Rom 16:17. [Again, the divisions and factionalism that Paul speaks about in these two verses, are those between Christians, not the world of secular politics.]
• αἵρεσις = a group that holds tenets distinctive to it, sect, group; ... (with negative connotation) 1 Cor 11;19, Gal 5:20. [This is condemned by Paul because he is discussing divisions in the unity of the faith.]
• ὅμοιος = things similar to the previous list, namely, "sexual immorality, impurity, and debauchery; idolatry and sorcery; hatred, discord, jealousy, and rage; rivalries, divisions, factions, and envy; drunkenness, orgies"

Paul and Jesus lay a great deal of emphasis on the "Unity of the Faith/Spirit" in many places:

• Eph 4:3 - being diligent to keep the unity of the Spirit in the bond of peace;
• Eph 4:13 - until we all may attain to the unity of the faith and of the knowledge of the Son of God, unto a complete man, to the measure of the stature of the fullness of Christ,
• Col 3:14 - And beyond all these, put on love, which is the bond of perfect unity.
• John 17:11 - And I am no longer in the world, and yet they are themselves in the world, and I am coming to You. Holy Father, keep them in Your name, which You have given Me, that they may be one as we are.
• John 17:23 - I in them, and You in Me--that they may be perfected in unity, so that the world may know that You sent Me and loved them even as You loved Me.
• Rom 15:5 - Now may the God of endurance and encouragement give you to be of the same mind with one another, according to Christ Jesus,

This is NOT to suggest that Christians should be uniform, but simply united in faith, spirit, mind, love and purpose (as documented above).

James restates Paul here, applying it to his own synagogue. He says that people are using the Roman justice system to sue one another, for selfish gain (rather that just asking God for their care):

[Jas 4:1-3 NASB20] [1] What is the source of quarrels and conflicts among you? Is the source not your pleasures that wage war in your body's parts? [2] You lust and do not have, [so] you commit murder. And you are envious and cannot obtain, [so] you fight and quarrel. You do not have because you do not ask. [3] You ask and do not receive, because you ask with the wrong motives, so that you may spend [what you request] on your pleasures.

But Paul's treatise on the lifelong struggle with the urges of their bodies is given in Romans 6-8. In this section, Paul personifies sin as "Mr. Sin," an evil alien that "has a mind of its own." So the saint who wants to do the right thing, often finds himself "owned" by "Mr. Sin inside my body parts." Paul says that the body places the saint in a wretched state, always at odds with "its" desires. Deliverance only comes through the transformation performed by God when the saint breaths his last. That is, what Jesus did was not to take away our bodies now, but to give Mr. Sin a death sentence, while the believer will be changed and live forever, free from Mr. Sin, in brand new bodies.

The translations have a good representation of the lexical meanings of the words. So, below are some passages from Paul's letters that give descriptions of the actions representing these terms.

Word translated διχοστασίαι:

I appeal to you, brothers, to watch out for those who cause divisions (διχοστασίας) and create obstacles contrary to the doctrine that you have been taught; avoid them. 18 For such persons do not serve our Lord Christ, but their own appetites, and by smooth talk and flattery they deceive the hearts of the naive. 19 For your obedience is known to all, so that I rejoice over you, but I want you to be wise as to what is good and innocent as to what is evil.

(Rom. 16:17–19, ESV)

Word translated factions: αἱρέσεις

17 But in the following instructions I do not commend you, because when you come together it is not for the better but for the worse. 18 For, in the first place, when you come together as a church, I hear that there are divisions (σχίσματα) among you. And I believe it in part, 19 for there must be factions (αἱρέσεις) among you in order that those who are genuine among you may be recognized. 20 When you come together, it is not the Lord’s supper that you eat. 21 For in eating, each one goes ahead with his own meal. One goes hungry, another gets drunk. 22 What! Do you not have houses to eat and drink in? Or do you despise the church of God and humiliate those who have nothing? What shall I say to you? Shall I commend you in this? No, I will not. (1 Cor. 11:17–22. ESV)

The parallel term σχίσματα:

I appeal to you, brothers, by the name of our Lord Jesus Christ, that all of you agree, and that there be no divisions (σχίσματα) among you, but that you be united in the same mind and the same judgment. 11 For it has been reported to me by Chloe’s people that there is quarreling among you, my brothers. 12 What I mean is that each one of you says, “I follow Paul,” or “I follow Apollos,” or “I follow Cephas,” or “I follow Christ.” 13 Is Christ divided? Was Paul crucified for you? Or were you baptized in the name of Paul? 14 I thank God that I baptized none of you except Crispus and Gaius, 15 so that no one may say that you were baptized in my name. 16 (I did baptize also the household of Stephanas. Beyond that, I do not know whether I baptized anyone else.) 17 For Christ did not send me to baptize but to preach the gospel, and not with words of eloquent wisdom, lest the cross of Christ be emptied of its power. (1 Cor. 1:10–17. ESV)

Paul gives the opposite characteristic from these in 1 Cor. 13.