All answers are only true for a campaign; the ACKS corebook describes alignment for the Auran Empire setting.
There’s some literary examples that are quite useful, mainly Poul Anderson (e.g. Three Hearts and Three Lions) and Michael Moorcock (all the Eternal Champion novels and stories).
Basically, someone who actively worships and advances the cause of the Lawful gods (patrons of humanity and demi-humanity) is Lawful; someone who does the same for the Chaotic gods (chthonic enemies of humanity) is Chaotic. In OD&D, a cleric could start out Neutral, but had to eventually choose between Law and Chaos.
Dungeon Crawl Classics has a great take on two-pole alignment, and makes Neutrality a viable active choice in itself. I recommend checking it out.
I’m not sure if this is in the ACKS book (might’ve been), but your example of the barbarian would be a character who eventually becomes Lawful by deed despite the lack of intent (unless he also sacks cities and brings empires to ruin).
The Good/Evil thing is a bit mixed. A Lawful character can absolutely be evil: maintaining order and civilization and advancing the cause of a realm while torturing your dissenters, assassinating your enemies, and ruthlessly amassing power and wealth for yourself (because you just know you are the one who knows best how to preserve the proper order!)…
Meanwhile, Chaotic creatures and characters are much more likely than Lawful ones to be evil, on balance; destroying civilization tends to involve doing a lot of bad things to a lot of people! But there can also be creatures who are more just Chaotic by nature (Anderson’s elves and trolls, elves and magic-users in Lamentations of the Flame Princess, etc.), and could actually act honorably. The Fae of medieval and renaissance legend are a good example of a Chaotic folk: not always outright evil, but capricious, willful, and often amoral.
I usually specify the Law/Chaos axis for each setting and campaign.
In my conception of R. E. Howard’s Hyborian Age, magic (or most of it), the chtonic “gods” and monsters, the worship of Set, etc. is all Chaotic; being Lawful is defined by fighting against those things. Human civilization is less important, although champions of civilization like King Conan and King Kull will inevitably find themselves in opposition to the forces of Chaos.
In my Crimson Sun campaign, Chaos is pretty much defined by the plundering of the world’s life-energy to fuel arcane magic, and all who do that (or who follow the Sorcerer-Kings who are the greatest plunderers of all) are Chaotic. Those who oppose them (including druids and the Veiled Society, etc.) are Lawful. Most people don’t dare oppose them, but don’t have the option to join them, so they’re Neutral.
In my Land of Kings setting (based on Ancient Mesopotamian mythology), Tiamat the Ummu-hubur is Chaos, the primordial saltwater ocean; Apsu the Cosmic Waters is Law, the primordial freshwater ocean (aquifers). The Young Gods who took Apsu’s power and reside in the Apsu Waters are Lawful, and the very concept of human civilization they ultimately created is Lawful; the Chaotic monsters, beastmen and cultists who seek to overthrow civilization are all ultimately followers of Tiamat. (I’m a bit undecided on the Underworld Gods; probably Neutral, maybe Chaotic, maybe a mix of both.) Humans ally with one or the other by their choice of worship, but only priests, leaders, and champions are really Lawful or Chaotic.
In my as-yet-unnamed original setting, Law and Chaos are (for the humans of the campaign area, who are the primary frame of reference) defined by the two deities, Light and Dark, the Creative Urge and the Destructive Urge (comparable both to Eros and Thanatos of psychoanalytic theory and the Spenta Mainyu and Angra Mainyu of Zoroastrianism). Neither side is evil (although the Universal Church of Light holds that the Dark is), although a hefty portion of the Dark cults (all of which are illegal in the lands dominated by the Church) are nihilistic to the point that they encourage or embrace amoral or downright evil behavior; but at the same time, plenty of priests of Light are corrupt and venal, some even downright twisted and evil. Law and Chaos have fairly little meaning for non-cleric PCs, though, partly because there is no concrete or independently active divinity in the setting, just systems of belief.
The last bit was all written around the central concept of “I want clerics and anti-clerics” (to accommodate classic D&D villains like Lareth in T1, and the “EHP” - Evil High Priest), bundled with “but I want venal and corrupt priests, too.”
If you can be bothered to wade through old Grognardia posts, especially regarding the Dwimmermount campaign, you can find a good deal of very old-school thinking on Law and Chaos that I like.