Communism isn’t “people working for the common good,” it’s people working to improve their own material conditions.
Same goes for capitalism. Why is it called communism then, if your definition doesn’t even contain any reference to anything communal? At the very least, it would have to be “people working together to improve their own material conditions”, but that’s perfectly acceptable in capitalism as well.
Come on now, if you want to have a debate about this, at least try to make argument that doesn’t fall apart at the slightest breeze.