This is all explained in AC120-28D !!!
This is all well explained in FAA AC120-28D (issued 13 June , 1999).
See section 4.3 of the AC in particular, reference use of an AH vs. a DH, for Fail-Op versus Fail-Passive. It also matters whether with or without rollout capability (e.g., note the original triple channel SPZ-1 B747s had FAIL-OP capability, but NO rollout system even installed [thus the Op-Spec visual confirmation requirement before TD], ...until later rollout systems were added via the "rollout special condition" which was used for their initial certifications.
Derek Helmore (UK CAA inspector) and I (for FAA) first sorted out the use of Fail Op and Fail Passive systems for Cat III, with Fail Op using only an alert height [and no need any more for a (low) DH] many decades ago, back when we were negotiating TWA's L1011 minima into the UK, ...versus Concorde minima to be used into KJFK and KIAD... as well as when we later wrote the first drafts of ICAO Doc 9365AN/910 (ICAO Manual of All Wx Ops).
Further we (the FAA/JAA AWO HWG) addressed addressed these subjects again extensively, over the decade of the '90s, before and leading up to issuing the [somewhat] harmonized revised versions of AC120-28D and AC120-29A, when they were published. This was also the period when we [the AWO HWG] globally dropped any further used of the concept of MABH.
Bottom line is that [under US rules] for LAND3 mode [i.e., Fail Op], an alert height is typically used [usually set at either 100' HAT or 50' HAT], and no DA(H) is necessary. For Fail passive systems and ops [i.e., LAND2 or equivalent] a DH is applied (with some minor exceptions for certain specific irregular terrain airports where an IM or DA may still apply), primarily to accommodate the possibility of a fail passive AP disconnect.
A minor special case has evolved recently, using AC120-28D (section 5.8) provisions for a "Hybrid System", by combining use of an autoland system, monitored with a suitable AIII mode capable HUD. Per the AC, this combination can theoretically be considered as a Fail-Op system, and used with an AH [and without a DH] if AP LAND 3 mode is used as baseline, ...or used with a low DH, if the AP mode used is LAND 2 or equivalent. In either case, the AP is backed up by a suitable AIII capable HUD, for the [contingency] continuation case, or balked landing GA case.