FMO notation.

In general, the following notation is preferred: FMOn-X, where X is wave function and n denotes the many-body expansion (n is 2 or 3). Other extensions, such as PCM are put after a slash, such as FMO-RHF/PCM or simply FMO/PCM if RHF is understood or any wave function is meant. In principle, the PCM in conjunction with FMO also uses the FMO expansion of the electron density (specifically for solvent dependent calculations), therefore, the most pedantically correct usage would be as in FMO3-CC/FMO2-PCM. This strikes me personally as redundant and instead I choose to use FMO3-CC/PCM[2].

MFMO is used sometimes as an abbreviation of multilayer FMO. When appropriate, it is better to use the standard abbreviation, which is
FMOn-X1:X2:X3 (etc), where X1,X2,X3 etc are the wave functions. E.g., FMO2-RHF:MP2 has RHF for the lower layer (which includes all fragments) and MP2 for the higher layer (which has some fragments).

Why is PIEDA not called FMO-PIEDA?
This is because PIEDA was developed as a pair interaction energy for various pair interactions, not just those in FMO. In particular, the free state analysis is strictly speaking not FMO, or at least, not the traditional FMO. In other words, PIEDA can be applied to pair interactions in both FMO and other cases, so it is more general than just FMO. However, it is equally possible to broaden the FMO method to include an arbitrary reference, in which case PIEDA can be thought of as FMO2-based EDA. So, one may loosely refer to PIEDA as FMO2-EDA (since EDA interaction components are obtained at the two-body level).