From then onward titles like Final Fantasy (1987), Meridian 59 (1996), Pokémon Red & Green (1996), and a plethora of other single-player and multiplayer RPGs began putting their own twist on critical hits. Some games even let you build your character around improving their crit chances, like Rogues in the original World of Warcraft (2004). (Inverse)

There are two general methods of handling critical hits: In the first method, they simply do extra damage, usually multiplying the base damage by some number. In the second method, random results are generated from a "table" of possible effects, which range from extra damage to Subsystem Damage to instant death. (TV Tropes)

A higher haste value doesn't necessarily have to reflect a higher amount of critical hits. Many games use two different stats for calculations - base stats, and modifiers. (D&D is a great example). Why not calculate critical hits based on the base of the haste stat, but still reflect the critical chance? (Game Development Stack Exchange)