The Ability "Organisation Lore: Organisation" is merely the knowledge of that organisation (see ArM5, page 66). It's what people mean when they say "institutional knowledge". The organisation could be a merchant guild, a monastic order, the church, the Order of Hermes, whatever. That an organisation has a lore does not imply anything supernatural.
However, some organisations are Mystery Cults. In this case, knowing about the organisation includes knowing about that organisation's mystery; hence why it helps with Mystery initiation. However, just because an organisation has a lore it doesn't mean that organisation is a Mystery Cult (or that it can become a Mystery Cult).
Finally, there are also some things that share (roughly) the game mechanics for Mystery initiation. This doesn't mean those actually are Mystery initiations, or that the organisation in question is a Mystery Cult. It is just for the convenience of the players that the game mechanic is duplicated.
Similar game mechanics don't imply any similarity between the in-character actions. For example, "Hunting a Stag" is an Ability roll made with the Hunt Ability, and "Preparing a Horoscope" is an Ability roll made with the Artes Liberales Ability. The similarity of the game mechanics aren't meant to imply that hunting and artes liberales are somehow similar.