What are accurate flight simulation models called? Is there a standard model?
14 CFR Part 60 regulates what constitutes a Flight Simulator Training Device (FSTD) for a particular aircraft model. FSTD is divided into two broad categories: Full Flight Simulators (FFS) and Flight Training Devices (FTD). A big difference between the two categories is FFS requires having motion cues, whereas motion cue is optional on FTD. Furthermore, the required accuracy of simulated flight dynamics (compared to flight test results) is much more stringent for FFS than is for FTD.

Each FFS and FTD is further divided into qualification levels (Level A to D for FFS, Level 4 to 6 for fixed-wing FTD). The higher the level, the more fidelity the simulator has to the actual aircraft. You may only get credit for Part 61 if you get training on the simulators with the appropriate levels for the systems/flight phases you are working with. 

The simulator manufacturer must submit a Qualification Test Guide to the FAA to demonstrate that the simulator is good enough on aerodynamic model, flight deck, visual system, motion cue, sound system for the appropriate level.

Anyone can make a flight simulator, to any arbitrary level of fidelity; but making a flight simulator good enough for the FAA is hard and expensive. Read this for a more in-depth but lay-man explanation of the different qualification levels.