What term should I use when I am quantifying the amount of matter, mass or weight?
Because of the reasoning provided by you, i.e. mass is universal, and weight is the product of mass and magnitude of the local gravitational acceleration, I suggest to stay at mass.  (A purist's view, taking @Ivan's comment into account.)

In addition, referring to the International System of Units, their brochure not only mentions the basic units, but the relevant section 2.1.1.2 starts with: 

Unit of mass (kilogram)