For the context you're describing:
A system is a collection of things that work together to accomplish a purpose (see M-W; #1d).
A system can be any size, from microscopic to astronomical. A system can be located in a building, or even be a network of things, some of which are contained in buildings, but typically, the building isn't an integral part of the system. There are exceptions, or a system can be integrated into a building, but usually, the building is just a place where the system is located, and might protect it from the elements if the building's purpose is specifically to house the system.
A plant usually refers to a building or collection of buildings at a location, the associated land, and the equipment there, which are employed in carrying on a trade or an industrial business or producing a certain "product" (which could be power). A plant may contain one or more systems or could be part of a network composing a system. (see M-W; noun #2).
If you're talking about a small device or system installed on someone's property, that would typically be referred to as a "system", "device", "machine", or a term for the specific kind of thing it is, rather than a "plant".
However, "a term for the specific kind of thing it is" complicates things. A "plant" can be any size, there isn't a specific size threshold below which something can't be called a "plant'. A power generation facility is referred to as a "power plant".
For roughly the last half century, there have been miniaturized power generation modules that serve the function of a power plant in remote areas. They have been referred to as a "power plant". The term has also been applied to the power generation module for things like space craft or landers (well, that's probably covered by "remote"). It isn't normally applied to something like a battery, that just stores power, but to something that creates power.
So if a power generation module is a "power plant" in a remote area, there's precedent for calling it the same thing if you stick it in someone's backyard.