Massive particle is indeed any particle with mass.
The different paths in spacetime followed by massive and massless particles will be visible to us as the particles following different paths through space.
As cavediver said, this is what makes geometry with a time dimension very different to purely spatial geometry. When all the dimensions are spatial there is only one type of path, our normal intuitive idea. When there is a time dimensions there are three separate types of paths.
Edited by Son Goku, : Small addition.
™ Version 4.2