You have most of the answer listed. A "fan coil unit" is a generic name for a box that contains a "fan" and a "coil". As listed in your answer, the coil could be designed to accept either cold or hot water. The fan would blow air from the room across the coil and either heat or cool it.
The coil could also contain refrigerant, such as in the case of a heat pump water heater. The same process would involve blowing room air across the coil to either heat or cool it.
When we say "fan coil unit" we usually imply that it was set up for a limited area such as a single room or maybe two. In most homes, we have a furnace with its blower and we place the coil above that to distribute cold air (or hot in the case of a heat pump) throughout the entire house. In that case, we call it an "air handling unit" because of its taking care of a much larger space.