As always, with Japanese, it's nuance, not hard-and-fast.

RYU can indicate a tradition that is generational (or if one is the founder, one has that intention. For example, Itto-ryu, a martial art founded with the intention of transmission, directly, person-to-person in a descending (and spreading) lineage.
RYU can also mean 'style.' Were someone to use the former meaning in referring to Iwama-ryu, that would be a declaration of independence or a revolt, However, the nuance here is that the difference is definitive enough to be notable. For example, one might refer to Endo-ryu or Tada-ryu (and maybe some do), but use of the term indicates that there is more of a separation--socially, stylistically or structurally --than merely a different way of moving within an umbrella organization.

HA - no one asked about this term - but this indicates a branching off that is different enough that it truly has changed - yet, it retains enough of the original that it's not a clear separation - ie., Ono-ha Itto-ryu, Chuya-ha Itto-ryu, Mizoguchi-ha, Itto-ryu

KAI just means an organization - a grouping, so to speak.
KAN is much the same, but it has the nuance of an edifice. If there was an "Aikikan," one would immediately imagine an imposing building as a headquarters.

Ellis Amdur