That's a great question. The answer is, the stuff you are reading in this thread is not right (you figured it out). The real version of the story is, there is this thing called the "Christoffel symbol," which tells you where, at every point in space, you would end up if you went in a certain direction, including which way you would be facing if you went that way. It relates three vectors: your direction of motion, the direction you are currently facing, and the delta to your direction of facing that would result from taking that direction of motion.

If you let your current momentum be your direction of facing, and let the same momentum also specify your direction of motion, the Christoffel symbol tells you what your momentum vector would be after an infinitesimal amount of motion. This can be integrated to find the version of a straight line appropriate for a curved surface (imagine an ant walking straight forwards on the surface of a cone or something), a geodesic. A changing momentum is like a force is acting, so that's gravity.

There is more to learn than that, of course. Many many many books have been written about general relativity and you can read them.