Is HVAC connection coding universal?
It is a de facto quasi-standard situation

The situation with thermostat wiring nomenclature is more or less the third case in your question: manufacturers cooperate to follow similar (albeit not identical) schemes for marking the terminals on/used with "North American style" 24VAC thermostats.  While the basic terminals are more or less universal, there is more variation in the terminals used for more exotic functions (such as multiple fan stages from the 'stat, heat pump reversing control, or humidity management).

However, there is a push in new system designs (led by mini-splits, modulating furnaces, and inverter modulating air conditioners/heat pumps) to not support this "standard" at all; instead, these systems use a proprietary communcations bus (similar in concept to an industrial fieldbus, but not based on any standard fieldbus or building-automation protocol) between the thermostat, the air handler/furnace, and/or outdoor unit.

Some hydronic systems (mostly in the European market) do use the OpenTherm standard, but due to the apparent lack of support for fan coils and air handlers in the standard, it does not see much use in North America to my understanding.  Larger commercial systems in the US are also more standardized as well, sometimes using BACNet or a similar standardized protocol for communications between the various system components.