A truly detailed discussion of the certification standards for FAA Flight Simulation Training devices simply isn't possible on a website like this. That's because FTDs are covered with FFSs under 14 CFR Part 60, which is 378 pages and around 270,000 words.
But importantly that information wouldn't be really useful to you. Where as BATDs and AATDs are covered in a relatively terse Advisory Circular1 in terms of required controls and instruments, and a subjective evaluation, FTDs are defined in extreme detail with objective evaluation.
ATDs are evaluated at the request of the manufacturer, they are "products" which bare a regulatory approval and can be bought and used within regulatory standards without as much concern from the operator. FTDs on the other hand are evaluated at the request of the operator and if approved may only be used as described in the approval. If the simulator is moved, used in a different way, changes users etc. it will lose its approval.
Flight Training Devices come in different levels. Often it is cited as Levels 1 through 7. However in practice the FAA no longer issues Level 1–3 certifications, and Level 7 only applies to helicopters. That said the criteria laid out in FAR 60 for a Level 5 airplanes FTD is in a different appendix, and thus technically different from a Level 5 helicopter FTD.
One of the other primary differences, is FTD evaluation is much more objective. FTDs are required to have laid our a performance specification, and then prove they meet those specifications. An FTD is also required to be able to demonstrate that it can complete certain tasks per a specification.
All newly certified Level 5–7 FTDs would certainly qualify as what we refer to on this website as high fidelity complex simulators, and are going to be very capable.
When building a custom high fidelity complex flight simulator it can often be a good strategy to design it to meet a high FTD certification level, even an AATD certification would meet the regulatory requirements for it's use. This is because insurance companies may provide benefitial treatment to a Level 5 FTD over an AATD, and it is possible for the FAA to approve an FTD for uses beyond or at least different than the approved use of an AATD.