OEM means generic. If they don't have the 2 week turn around time you can be damn sure it's generic as that bullshittery is imposed by Core.
Also if they don't ask you to disable Find my iPhone that is also a sure sign of it being a generic.
That being said just get the generic, unless you are in warranty. Will make no difference at all to your life.
This is not true.There are genuine OEM batteries you can get. Apple’s Original batteries are manufactured by different Chinese companies,Huizhou Desay Battery Co. Ltd being one of them. You can bet some of those exact batteries end up on the market just missing the apple logo sign. Also Find My iPhone has got nothing to do with changing a battery on an iPhone. The problem is most of the repair shops use the word OEM loosely and lie a lot to people.