My most played supports in order are Leona, Morgana, Lux, Annie, Soraka and Janna.
I think Annie is the only one that is an obvious off-meta support. Some might say Lux is, but I play Lux as a mix of Morgana/Janna. Since both of those are easily recognized as viable supports, I don't think I require explaining the strengths and weaknesses of a Lux Support.
Annie is a very pocket pick. Situation has to be right for me to go with her.
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Body guard Ability - Depends on the intelligence of your opponent. She can stun one opponent at early levels, but won't prevent a Pre 6 All in. It's actually quite garbage. Synergy best with an ADC that can farm safely or has decent escapes.
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Engages/Disengage/Peel - Devastating Level 6 Engage against other AP and/or Utility Mage Supports. Good Level 6 Disengage against Tank Engage Supports. Honestly, after level 6, this is what you brought Annie for. I regard her as a pocket pick to negate Divers and Assassins. I want the enemy Diver to have to prioritize me for any glimmer of hope of being useful that game. Think of it in terms of how the Soraka pick changes priorities.
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Buffs - None. I've been hoping forever that Riot would allow Annie to place her Sunfire Cape spell on an ally. It's good defensive stats and has amusing synergy with Rylai (which is pretty core on Annie Support).
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CC - You have to buy it. Your Stun will not be enough for the game. I always pick up Rylai and sometimes pick up IBG if I feel like subjecting the enemy team to a tank version of Annie.
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Procs - Depends on your laning opponent. Annie can technically zone a skittish ADC from getting their CS, but that should not be happening pre 6. I've ran Thief on her if up against a Tank Support, since that's free gold. A lot of Utility supports are weak at harass, so Thief is good there. Almost all AP mages will bully you hard pre 6, so Coin is smarter.
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Waveclear - Depends on how much damage you are building. But don't. Seriously don't. That should not be why you picked Annie. If the enemy teams sees you clearing a wave, they know they can go ham on your team without fear of your disengage. Tank versions of Annie can pick up ZRot Portal if the team desperately needs wave management.
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ADC Synergy - Since Annie should be useless pre 6, it will be mostly long range ADC like Caitlyn/Ashe or others that have peel and escapes like Ezreal/Lucian/Graves/Tristana/Jinx/Quinn/Vayne. The thing is, you can't really bully your lane since Annie has nothing pre 6. That one stun will not stop a Yi running at you from river if you have overextended. It really won't stop anything. So when I say long range, I'm meaning an ADC that can farm safely instead of their usual bully tactics.
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Champions they counter - Annie tends to be my answer for Janna/Soraka lanes. I also purposely pocket pick her when I see something like Yi, Katarina, Xin Zhao or Wukong. Basically divers that have the habit of charging into a team with a legitimate chance of pulling off a 1v5. I pick Annie with the explicit purpose of stopping that.
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Countered By - Pretty much every AP Support Mage Pre 6. While Annie is lethal at level 6, it's a grueling struggle to get to that point when facing off against a Sona, Karma, Brand, [insert entire AP Support Mage roster here].
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Favorite Strategy - I like to have Annie with a sweeper hiding in an unwarded bush near her team. It seems like every game I pocket pick her, the Diver in question thinks it safe to jump on my team. I've actually landed a 4 man Tibbers stun against a team lead by an Ekko that thought they could win a 4v3 mid lane but failed to remember the Annie. They will remember the Annie. Also, being in charge of the vision game puts Annie in a lot of Over the Wall Tibber Stun opportunities. Protobelt Tibbers a three man waiting in a warded bush. Oh the joys of being Low ELO where half the players think vision is unimportant.