We’re using the term “straight-line” to avoid the more esoteric “geodesic”. Objects in free-fall (say, the Earth in orbit around the Sun) follow geodesics in spacetime, which can be alternately defined as paths which extremize the Hilbert-Palatini action (or one of the classical-mechanically equivalent actions) or as a path which parallel-transports its own 4-velocity vector. It’s perfectly unambiguous given the ontological setup of GR, which (again) is the framework the OP is implicitly asking about.
It’s important to understand that we’re talking about paths through spacetime, not just through space. Suppose, for instance, that I want to travel from right here, right now to a point 2 AU away from here, on the opposite side of the Sun, and six months from now (note that I had to specify both endpoints in both space and time). One path I could take is to go very quickly, perhaps even at the speed of light, straight through the Sun, and then stop at the other side and wait there for six months. But this is a very un-straight path: When I come to a halt and wait, I’m making a very sharp corner there. On the other hand, I could follow a semicircular arc, travelling through space at a constant speed, just like the Earth actually does move in its orbit (yes, I know that the orbit is an ellipse and not a circle; I’m approximating here). That path is, in some sense, “straight”, and it’s the path an object in orbit follows.