Battlefield control, scoring, and the final point

Points come from battlefields, and a battlefield pays out through three separate ideas: control, scoring, and winning. Most arguments come from running two of them together.

Worked example. On your turn you move a unit to an empty battlefield, Treasure Hoard, that you do not control. Moving there applies Contested (rule 190.3.a and rule 190.3.a.1), so at the next cleanup a showdown opens. Nobody else has units there, so you establish control and, because you have not scored Treasure Hoard this turn, that is a Conquer: you gain a point and its "when you conquer here" ability triggers (rule 348.2.a, rule 471.1 and rule 473.2). If an opponent later takes the battlefield and you take it back the same turn, nothing more happens: a player may score a battlefield only once per turn, by either method (rule 472 and rule 473.2.c). On your next turn, keeping control through your Beginning Phase is a Hold, worth another point and triggering hold abilities instead (rule 471.2).

Control. Control is established by having units at a battlefield at the end of a showdown or combat (rule 190.4). Outside combat you keep it as long as you have units there (rule 190.4.a). With no units there and the turn in an Open State you lose it in the following cleanup, unless a combat or showdown is ongoing there (rule 190.4.c and rule 323.5). While items are on the chain the turn is Closed, which is why you can kill your only unit there as a cost and still play its replacement to that battlefield. A unit that moves in where its controller already has control does not apply Contested (rule 190.3.a.1).

Two meanings of "score". Scoring a battlefield is the process of seizing or holding one you have not scored this turn, and it may or may not gain a point (rule 470 and rule 470.1). Card text that says "score X points", as Tryndamere, Barbarian and Ahri, Alluring do, means gaining points, not Scoring (rule 470.2 and rule 470.3). Tianna Crownguard was given errata because "score" muddled the two: her text now says opponents "can't gain points". That stops the point but not the Conquer or the Hold: the conquer ability still triggers, and the same holds when a point is replaced (rule 383.4.c.2.c and rule 383.4.d.2.c).

The Final Point. When a player tries to gain a point through a Conquer while one point from the Victory Score or higher, they gain it only if they have scored every battlefield this turn. Otherwise they draw a card instead (rule 473.1.b and rule 473.1.b.1). Points from anything other than a Conquer, such as a Hold or a spell, are not bound by this (rule 473.1.a.1).

Winning. A player wins if, in a cleanup, they have points greater than or equal to the Victory Score, 8 by default, and more points than any other player (rule 194.2 and rule 194.3). A cleanup cannot start while a chain item is resolving, so the win comes after the item has finished (rule 321). Tied at the Victory Score, play continues until one player has more in a cleanup (rule 194.2.b). Modes and cards, such as Aspirant's Climb, can change the Victory Score (rule 194.3.a). The one exception is repeated Burn Out, which wins at once (rule 432.3.c.1).

Battlefield abilities. Control of a battlefield is control of its abilities, and the controller adds them to the chain and makes the choices (rule 190.6 and rule 190.6.a). If nobody controls it and the ability names no player, the Turn Player takes responsibility for it (rule 190.6.b). If the ability names a player who makes a choice, that player controls it whoever controls the battlefield (rule 190.6.c).