First-Instinct Chains are chaining styles that require little thought or input to execute and are thus commonly used by newbies. Think of it as Puyo Puyo's equivalent of button-mashing. Without exposure to skilled players, internet resources, or actual Puyo games, new players sometimes look toward the games' AIs for guidance.
Harpy Stacking (ハーピー積み) refers to immediately filling up the first and last column, and then hoping your following Puyo randomly chain somehow with the Puyo on the side. This is similar to how Harpy and Yu (& Rei) play in the Puyo Puyo games. Jay and Elle also use this strategy in Puyo Puyo Tetris.
Harpy's version is to move one pair to the left, then one to the right. Yu instead does not move to the right until she has filled the left column.
Frog Stacking (カエル積み), also Nohoho Stacking (のほほ積み), refers to throwing down every piece onto the right side of the board and hoping for a random chain. This is similar to how Nohoho and Dongurigaeru, both frog characters, play in the Puyo Puyo games. Ai in Puyo Puyo Tetris also uses this chaining AI.
Tara Stacking (タラ積み) refers to placing your pieces so that they fill the first four rows. May possibly refer to the way Suketoudara plays. MEIKO in Puyo Puyo 39! (Hatsune Miku Project Mirai 2) also uses this chaining AI.