Aquatic
Dragonkin – Purple Puffer, Tiny Red Carp
Elemental – Gahz’rooki, Mr. Chilly, Pengu, Puddle Terror, Spawn of G’nathus, Tideskipper, Tundra Penguin
Flying – Mr. Chilly, Pengu, Tundra Penguin (these three have identical abilities)
Humanoid – Puddle Terror, Tideskipper, Young Mutant Warturtle
Magic – Emerald Turtle, Ghostshell Crab, Moonshell Crab, Tiny Blue Carp
Mechanical – Hydraling
Undead – Chuck, Gulp Froglet, Muckbreath, Snarly, Toothy
Beast
Aquatic – Albino River Cal, Bucktooth Flapper, Celestial Calf, Flat-Tooth Calf, Mudback Calf, Periwinkle Calf, River Calf, Spiketail Beaver, Sumprush Rodent
Dragonkin – Ash’ana, Celestial Calf, Cinder Pup, Clouded Hedgehog, Dreadmaw, Silent Hedgehog
Flying – Dreadmaw, Lurking Owl Kitten, Silithid Hatchling, Tito, Wind Rider Cub (note: Dreadmaw's attack is Glowing Toxic which does not get an attack type bonus)
Magic – Ash’ana, Feline Familiar, Moon Moon, Thaumaturgical Pig, Vengeful Porcupette, Warp Stalker, Xu Fu
Mechanical – None!
Critter
Aquatic – Mudshell Conch, Rapana Whelk, Rusty Snail, Scooter the Snail, Shimmershell Snail, Silkbead Snail (so all snails sans Zoom)
Dragonkin – Fire Beetle, Lava Beetle, Malayan Quillrat, Malayan Quillrat Pup, Porcupette, Wolpertinger (the beetles and quillrats/porcupettes have identical abilities)
Elemental – Crispin, Fire Beetle, Imperial Silkworm, Lava Beetle, Zoom
Magic – Baneling, Mudshell Conch, Nether Roach, Nibbles, Rapana Whelk, Rusty Snail, Scooter the Snail, Shimmershell Snail, Silkbead Snail, Zergling (the Nether Roach is the only non-snail and non-collector’s edition pet in this category!)
Mechanical – None!
Dragonkin
Aquatic – Dream Whelping (doesn’t really count—the move is Toxic Skin)
Humanoid – Albino Chimeraling, Dark Whelping, Death Talon Whelpguard, Lil’ Deathwing, Nightmare Whelping, Soul of the Aspects (Albino Chimeraling and Dark Whelping have identical movesets)
Mechanical – Stormborne Whelping, Thundering Serpent Hatchling, Wild Golden Hatchling, Wild Jade Hatchling (all of these are Call Lightening and only the Stormborne Whelping has a unique moveset—it’s also probably the easiest get as it’s the only one that requires neither rep nor the BMAH!) **
Undead – None!
Elemental
Critter – Core Hound Pup, Jade Tentacle, Ruby Sapling, Tainted Waveling, Teldrassil Sproutling, Withers
Dragonkin – Ashstone Core, Cinder Kitten, Crimson Geode, Elementium Geode, Fel Flame, Nightmare Lasher, Ominous Flame, Searing Scorchling, Spirit of Summer Flying – Grinder, Living Sandling, Lumpy, Panderan Air Spirit, Pebble, Phoenix Hatchling, Tiny Twister (Grinder, Lumpy, and Pebble have identical movesets)
Humanoid – Crazy Carrot, Droplet of Y’Shaarj, Gooey Sha-ling, Sandling, Nightmare Lasher, Nightshade Sproutling, Pandaren Air Spirit, Terrible Turnip, Tiny Bog Beast, Winter Rageling (Nightmare Treant also has humonaid moves, but obviously that won’t help with this achievement! Igneous Flaming is not available until next Midsummer Festival, but it will eventually be in this category)
Mechanical – Grinder, Lumpy, Pebble, Tiny Twister (Grinder, Lumpy, and Pebble all have Quake, and I highly recommend them against Beast heavy teams)
Undead – Amethyst Shale Hatchling, Crimson Shale Hatchling, Emerald Shale Hatchling, Droplet of Y’Shaarj, Gooey Sha-ling, Ruby Droplet, Tiny Shale Hatchling, Topaz Shale Hatchling (all Shale Hatchlings have identical movesets)
Flying
Aquatic – Azure Crane Chick
Humanoid – Crow, Dread Hatchling, Fledgling Warden Owl, Gilnean Raven, Long-Eared Owl, Nightwatch Swooper, Northern Hawk Owl, Sky Fry (Fledgling Warden Owl/Long-Eared Owl/Nightwatch Swooper/Northern Hawk Owl have identical movesets. Sky Fry is the only Flying pet whose Humanoid move isn’t Darkness)
Humanoid
Aquatic – Crusher, Curious Oracle Hatchling, Murky, Ore Eater, Terky (Gurky/Lurky, only available in EU. Luckily, they share identical movesets with Murky and Terky) **
Dragonkin – Deathy, Murkablo (short of winning the lottery, I bet most of us will never have these pets. *cries into hands*)
Flying – Anubisath Idol, Harpy Youngling, Moonkin Hatchling, Qiraji Guardling
Mechanical – Anubisath Idol, Baby Winston, Crusher, Curious Wolvar Pup, Grunty, Ore Eater, Rotten Little Helper
Undead – Murkablo, Murkidan, Sister of Temptation, Sporeling Sprout, Wretched Servant
Magic
Dragonkin – Abyssius, Hungering Claw, Stardust
Flying – Enchanted Broom, Enchanted Pen, Jade Owl
Humanoid – Creeping Tentacle brought this number up to 11
Mechanical – Empowered Manafiend, Enchanted Broom, Minfernal, Netherspace Abyssal
Mechanical
Aquatic – Lifelike Toad
Critter – Iron Starlette, Lifelike Toad
Dragonkin – Mechanical Panderan Dragonling
Flying – Ancient Nest Guardian, Corrupted Nest Guardian, Darkmoon Zeppelin, Mechanical Chicken, Mechanical Pandaren Dragonling, Mechanopeep, Race MiniZep, Robo-Chick, Rocket Chicken (Mechanical Chicken/Robo-Chick have identical movesets)
Humanoid – Clock’em, Clockwork Gnome, Darkmoon Tonk, Draenei Micro Defender, Lifelike Mechanical Frostboar (the Micro Defender only Humonaid moves are the shields)
Magic – Lil’ Bling, Race MiniZep, Rascal-Bot, Stonegrinder, Sunreaver Micro-Sentry, Runeforged Servitor (Servitor's only Magic move is its shield)
Undead – Rabid Nut Varmint 5000
Undead
Aquatic – Blightbreath, Spirit Crab, Zomstrok (Bligthbreath's one Aquatic move is Acid Rain, so it doesn't hit hard and it has a three round cooldown, but this is only the second pet to have the Dreadful Breath/Acid Rain combo all-in-one, which is nice!)
Critter – Blighted Squirrel, Frostwolf Ghostpup, Ghastly Kid, Ghastly Rat, Ghost Maggot, Infected Fawn, Infected Squirrel, Stitched Pup
Dragonkin – Blightbreath, Fetish Shaman, Frosty, Scourged Whelping, Sen’jin Fetish, Voodoo Figurine
Elemental – Fetish Shaman, Fragment of Anger, Fragment of Suffering, Frosty, Fungal Abomination, Landro’s Lichling, Lil’ K.T., Mr. Bigglesworth, Sen’jin Fetish, Voodoo Figurine
Flying – Blighthawk, Bone Serpent, Son of Sethe, Vampiric Batling
Humanoid – Blightbreath, Bone Serpent, Cursed Birman, Fragment of Desire, Macabre Marionette, Scourged Whelping, Weebomination, Widget the Departed (Fragment of Desire’s only Humanoid moves are the shields)
Mechanical – Blightbreath, Boneshard, Son of Sethe
** has pets unavailable in either EU or US
It's definitely a good choice for a lot of people! Might put that in my original post as an option/suggestion.