Yes, it is a type of matter. To show that air is matter show that air occupies space and has mass, or "weight"., to do this weigh an empty balloon then inflate the balloon and weigh it again, this will demonstrate that air has mass and occupies space it is therefore matter.
Air is certainly matter; air has mass, has takes up volume. I am not sure that the definition you give is one scientists would use; we would more likely say matter has properties of mass, volume and can be detected by appropriate instruments.
By definition, anything 'filled with vacuum' would weigh nothing, as a vacuum means an area that contains no matter. Since the balloon would need matter in it to weigh anything, a vacuum balloon would be lighter than the heavier air around it, as air is made of matter.