Here is a prime example of a thread hack, yet connects the range of suggestions accumulated by OP's post. Apology in semi-logical order of details, more overlaps than a line of senators beachside in Miami. (shudder shudder)
The only reasonable justification of a self-centering 4-jaw is production work; the variety with master jaws switched out for tooling that holds a casting etc.
Only use encountered was turret lathes and stock serrated jaws running material, basically cube shape iron castings. The uncut base was faced whilst chucked on turned end in a 3 jaw; became a sealing flange with 4 bolt holes in corners. Terminology-wise that was more chucker than turret, individual parts vs bar feed.
FWIW, there are self centering 2 jaws, 3 jaws, 4 jaws, 6 jaws.
There are independent 3 jaws, and 4 jaws.
All available (with sufficient $$$.$$) in solid outside jaws, solid inside jaws, reversible jaws, and 2 piece jaws.
Here, they separate into plain jaws and stepped jaws. Plain jaws can be radiused for OD, or carry a narrow point, mainly accommodating wider range of diameters.
At this level, 2 piece jaws have about 3 patterns attaching top to master; slotted, keyed, 60° serrations.
Some models offer soft jaws and hardened jaws.
Those diverge once more into smooth jaws and serrated tooth jaws. Typically, serrated jaws are hardened, in generating incredible grip, they're marring the surface, an 1/8" of aluminum averts that.
On rare occasions, an independent 4 jaw will be run just opposing pair, fitted with tooling for really off centered needs.
By the time you get 16" swing, consider a crane. There are various lifting types, gantry's seem most practical.
At about 24" swing, starts another option. There are face plates with four bolt-on jaws for horizontal lathes, otherwise like vertical boring machines. Always hard, serrated, some reversible by removing and turning around. Used sets? Start around $2500.
For my $.¢, best choice is 2 piece jaws, supplied with reversible serrated hard jaws offers greatest flexibility. Whether 2-3-4 or 6 depends what machine I'll hang it on. If you acquire lathes, an operative search begins considering which variety of spindle design.
I buy used. A South Bend and Logan share threaded spindles, the real lathes D1-3 and D1-6. Easiest to mount, reversible rotation, most plentiful therefore most economical. Holding out on a rather exceptional 14", just because bolt on chuck; on faceplate, different than what's known as A-1 type.
The real 'trick' dialing in is visualizing what needs to happen movement wise of part, not indicator. Often, I start by NOT having the indicator touch all the way around. Makes the high side very apparent, especially on larger diameters. Mounted on cross slide, it's plenty fast to adjust contact to final it in.
Here are two well-knowns with impressive credentials. Watch all, it's entertaining, or skip to 4:50, then 8:50.
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