I'm afraid your opinion doesn't get to redefine well-known words like theocracy.
To be a theocracy, a realm must be ruled by the religion.
Despite dictionary definitions what constitutes a theocracy, particularly a BM theocracy, is a tricky question. None of the SA "theocracies" are not ruled by SA itself, rather they are ruled and guided by SA beliefs (a significant difference). Secular politics is very much out of the Church's control and there are occasional conflicts between secular and ecclesiastical authorities.
To put it another way, the SA theocracies are more like modern "Islamic republics" than the Vatican City; the government itself is separate from the religious institution but the realm's laws are still governed by the tenants of the religion, and its decisions are still guided by religious principles.
This would differ from a realm like the Libero Empire, which has a majority and influential Astroist population (I would not include Summerdale in this category as while there are significant Astroist nobles and temples they are neither the majority nor hugely influential) because while individual nobles might be influenced by Astroist beliefs the Empire's decisions and laws are not primarily guided by/based on the tenants of SA in a truly meaningful way.
