He knows he’d better define Capitalism because it’s a word that means too many things by default. He means specifically:
That economic condition in which there is a class of capitalists, roughly recognizable and relatively small, in whose possession so much of the wealth is concentrated as to necessitate a very large majority of the citizens serving those capitalists for a wage.
Crucially, Chesterton gets it right out of the way that Capitalism isn’t simply equivalent to “private property.” It drives me nuts when people use it that way because it’s not a useful or accurate definition; and as Chesterton is taking pains to argue, a simple "boolean allowance" of the existence of private property isn't enough to avoid very bad social outcomes.