You could consider a helicopter with a 2 bladed rotor to be similar to your balanced weight example, and you could do a web search for helicopter and gyroscopic effects on the rotor.
A solid disk, or a set of 3 or more weights/blades/whatever, has the same moment of inertia about every axis through the center of the disk. A pair weights has a large moment of inertia about one axis, and a small one about the perpendicular axis.
That difference may or may not be important, depending on the dynamics of the complete rotating system and its rotation speed.