The Plucky Squire

Despite the wording implying there are two different game modes to play, these are actually just two different difficulty settings. Adventure Mode is the harder of the two (relatively), and they stack, so playing on Adventure Mode will unlock both achievements.

Accessibility options don't negate this (or any other) achievement, so feel free to turn on Invincibility, One Hit Kill, and anything else to make this completion a piece of cake for almost its entirety. For whatever reason, invincibility isn't active during the Violet and Thrash boss fights where you play as them, so you're going to have to do them legit. They aren't that bad though and are mainly minigames as opposed to requiring combat skills.

As a side note, both of these achievements track, strangely. They go up 10% every time you finish a chapter; there are ten chapters in the game.