I'm curious where the FAA says that. I'm not sure why it would make a difference unless there's a LOC stepdown the GS will make you bust.
It is not said explicitly, but since circling uses an MDA and an MDA is only associated with a Non Precision Approach (NPA) procedure, the only NPA procedure on an "ILS or LOC" procedure is the LOC procedure. An ILS GS can be used as an advisory GS as long as any step downs and the MDA are complied with. A Circling Minimum Descent Altitude (CMDA) is always to an MDA, never a DA. So the only procedure on the chart that has an MDA or CMDA is the Localizer procedure. You will never see a CMDA on any FAA chart that provides "only" a vertically guided procedure option (ILS, LDA/GS, LPV, LNAV/VNAV). Also, a CMDA may not be lower than the highest Straight-In MDA when the approach procedure has a straight-in option.
Current Definitions from the PCG are:
MINIMUM DESCENT ALTITUDE (MDA)− The lowest altitude, expressed in feet above mean sea level, to
which descent is authorized on final approach or during circle-to-land maneuvering in execution of a standard
instrument approach procedure where no electronic glideslope is provided.
DECISION ALTITUDE (DA)− A specified altitude (mean sea level (MSL)) on an instrument approach
procedure (ILS, GLS, vertically guided RNAV) at which the pilot must decide whether to continue the approach
or initiate an immediate missed approach if the pilot does not see the required visual references.
NONPRECISION APPROACH PROCEDURE− A standard instrument approach procedure in which no
electronic glideslope is provided; e.g., VOR, TACAN, NDB, LOC, ASR, LDA, or SDF approaches.
PRECISION APPROACH PROCEDURE− A standard instrument approach procedure in which an electronic
glideslope or other type of glidepath is provided; e.g., ILS, PAR, and GLS.
APPROACH WITH VERTICAL GUIDANCE (APV)– A term used to describe RNAV approach procedures that
provide lateral and vertical guidance but do not meet the requirements to be considered a precision approach.