Please help me understand my HVAC wiring
Your air handlers don't need those connections (and aren't even equipped to use them)

The explanation for what's going on here goes back into the mists of history, back to when forced-air, fuel-fired furnaces and fan-coil air handlers were first developed.

Back in the bad old days

Early gas furnaces had only two thermostat wires, R and W.  A call for heat from the thermostat closed the gas valve and fed gas to the burner, which the pilot then ignited, producing heat.  (Oil burners are roughly similar save for having more combustion control complexity.)  This was fine in gravity duct systems, but when forced air came to pass, something had to turn the fan on and off at the right time.  You can't have the fan turn on with the burner or else you'll get a bunch of cold air out until the heat exchanger in the furnace warms up; likewise, turning the fan off immediately when the call for heat goes away isn't always wise either.  So, a fan thermostat (often combined with the high-limit in more modern furnaces) mounted in the furnace plenum is used to switch the fan on and off autonomously.

This means that when a G wire is added to such a system, it is a request for the blower to come on.  The blower can come on without a call for it from the thermostat, and in modern furnaces, it is possible for the furnace control to override the thermostat's blower call (say if a call for heat comes during a call for fan, to allow the heat exchanger to come up to temp as it should).  Adding A/C means that the furnace needs to know to turn the fan on during a cooling call, so the Y wire is also routed to the furnace control in these systems so that the fan can be run at the correct speed during a call for cooling.

Enter electric, stage right

However, some people didn't want to mess with gas in their house, and other folks wanted to get rid of messy oil tanks or expensive propane, and its attendant delivery hassles.  For them, the first alternative that developed was what is called an "electric furnace" -- this is a fan-based air handler with resistance heating elements in the airstream.  While not as economical as piped fuels, electric heat was seen as safer as it couldn't leak and/or explode, and also is cheaper than trucked-in propane or heating oil.

When the early electric central heating furnaces were designed, though, the designers had a decision to make: who should control the fan?  Unlike fuel-fired systems, with electric heat the full heating power is available instantly and the peak temperatures are happening inside the duct.  This means that the fan needs to come on basically instantly to keep temperatures within the air handler acceptable.  As a result, either the air handler or the thermostat could get delegated the job, and to save some money on a fan relay in the air handler, the job got left to the thermostat.

Hydronic fan coils like yours and direct-expansion heat pump air handlers are the same way -- hydronic heat is available whenever the boiler and circulator are going, and likewise with a heat pump where heat's on whenever the reversing valve is set for heat, the compressor is going, and the outdoor ambient isn't too cold.  In this scheme, the G wire is a demand for fan from the thermostat -- if the thermostat calls for heat but not blower, then the air handler will shut down on high-limit after a little while to protect itself from damage.  Likewise, the Y wire is not needed by the air handler since the thermostat will call for blower along with a call for cooling.

Fast-forward

Since HVAC supply houses don't want to stock 57 flavors of thermostat, nor do HVAC people want to have to sort through thermostats on their truck just to find a compatible one, modern thermostats are compatible with a wide variety of systems: gas or oil forced-air furnaces, electric forced-air furnaces, hydronic zone valves or circulators controlling fan coils, heat pumps, or even mixed systems.  Hence, all modern thermostats have a setting for fan control -- older ones used a jumper or switch inside the thermostat, while newer 'stats like yours set this mode using a software option.  In the "gas/oil" or "fossil" position or mode, the thermostat only calls for blower if the fan switch is set to ON.  In the "electric" position or mode, the thermostat calls for blower if the fan switch is set to ON, or if the fan switch is set to AUTO and a call for heating or cooling is taking place.

In your case

Your fan coils are designed along the paradigm of an electric furnace; namely, that the thermostat is in charge of the G wire.  In fact, neither Y nor W are even connected to any circuitry within your air handlers -- the terminals are provided there to serve as convenient wire-landing-points, that's all.

So, both your thermostats are set up correctly, and the 0180 setting should not be changed or else you'll get quite poor performance out of your system unless you select the fan ON all the time (albeit not a high-limit trip since hydronic fan coil units can't meltdown due to lack of airflow, unlike their electric or fuel-fired counterparts which need high-limit protection for safety's sake).

As to the C wires going back to the zone valves? That's just a return path for electricity so that you have a closed circuit from the air handler transformer, to the thermostat, to the zone valves, and back to the transformer.  (It's also where the C wires coming back from the air conditioner compressor contactors join in.)