Thanks a lot for your question. It is a very good one. I will give one answer, and one comment. I hope this will clarify the situation.
You are right when you say that matter is massive. This is indeed how it is defined (for instance in the dictionary). Now, at the most fundamental level we find atoms and their constituents. The latter comprise quarks and electrons, as detailed in this post. All those elementary particles are massive (although quarks and electrons only contribute to a small fraction of the masses of atoms; let's however discuss this specific point later if you are interested in it, as this is only weakly connected to the question).
In contrast, even if neutrinos are said to belong to the 'matter sector' of the Standard Model, they are not components of atoms. They appear in a variety of processes and then escape freely. For this reason, they can be either massive or massless. It is thus more a matter of vocabulary to call them 'matter particles'.
Finally, note that mass is an intrinsic property of an object (arising from its constituents in the case of a composite object). On the other hand, weight is a force that has for origin gravity. For instance, on Earth weight will be the effect of the gravitational field of the planet on a mass. The same mass will hence have a different weight on the Moon.
Again, thanks for passing by and happy holiday!