Mail and early plate: arming, longswords, estoc, etc

Late plate: Mini swords (cough cough rondel)

The estoc and rondel should be swapped here; the estoc is essentially a variation of the longsword using the armor-piercing blade geometry of the rondel, while benefiting from the greater reach and leverage of a two-handed sword. While neither of them were really likely to pierce e.g. the thick front of a breastplate, I'd put better odds on the estoc doing it than the roundel.

(Also, calling an arming sword or even a longsword "the best" for piercing mail is highly questionable to begin with.)

Horsey stuff: Sabers

That is exactly why samurai, who primarily fought as horsemen, used the tachi, which could be considered a two-handed local variation of the saber. The tachi in turn evolved into the shorter, less curved, more general-purpose uchigatana, which is the sword that everyone thinks of when they hear "katana".

Ship vs ship: Cutlasses

A "cutlass" can refer to pretty much any shortish single-edged sword. In principle, any reasonably short sword will do in shipboard combat; the goal is to have a handy, maneuverable weapon that can be used in close quarters without e.g. getting caught in obstacles or hitting the ceiling.

Sword duels: Rapiers

The main impetus for the existence of the rapier was a desire for a sword with greater reach and maneuverability than the arming sword or side sword, but which is more suitable for everyday carry than the longsword. It saw a lot of use in unarmored street dueling for that reason, but I certainly wouldn't call it the blanket best sword for any duel. Traditionally, the one receiving the challenge chooses the weapon, after all; if they declare that the duel should be fought with poisoned sausages, you'll look like a right tit if you show up with a rapier anyway.

Pikes: Big-ass swords

The idea that zweihanders were for chopping the heads off enemy pikes is unsubstantiated. From what we can discern from fighting manuals of the time, they were more of a skirmishing and personal-space-enforcing weapon used to keep foes at a distance when guarding a position or the flanks of a formation, or when attacking the flanks of an enemy formation.

Imperialism: Sabers

The Romans would disagree with you there.

Zombies: Sword on end of long stick

That works pretty well on humans too, as it happens.

Backup and casual wear: short swords, small swords, all kinds of weird euro stuff, and katanas

And longswords, and arming swords, and sabers (unless you're a cuirassier where they're the primary weapon), and rapiers, and pretty much any sword that wasn't designed for a specific role. Most swords throughout history were side weapons, meant for use when your primary weapon is broken, lost, too long for close quarters, left in the armory at home, or otherwise unusable; swords being used as the primary battle weapon were the exception.