Type C (the "europlug") is a compatibility low-amperage plug (2.5A @ 250V) for shavers and small lamps and USB power adapters and such. It was designed to be compatible with the high-power Type C, E, F, and K popular in mainland europe, but providing a cheaper plug, and allowing for dedicated low-amperage circuits (e.g. shaver sockets which would be safer in bathrooms).
Type H (and N, which is brazilian) were then designed to be compatible with the Europlug in turn.
> Then, main room air conditioners use Type M - but at least that's standardized...except for when it isn't.
Possibly rather than Type M (which is a 15A plug) it's the 30A plug from the same over-arching BS546 standard? Or a similar-looking 3-phase plug? That's what I'd expect for high-power applications like AC. I assume the standard Type H is 15 or 16A single-phase aka the usual mains.
The M used in Israel might be different than the one pictured as it is used exclusively for 3 phase lines (so an AC that doesn't need 3 phase can then have a regular plug)