The bowling ball analogy has been rightly criticized. Too many people get the wrong idea, mostly by concentrating on the irrelevant details of the analogy. You point out one possible wrong conception. That's why professionals prefer to stick to the equations. At a simple level, good for most things, we have the force equation from Newton. This is good enough for things on Earth, or even for orbits of binary stars around each other. Even for most orbits around black holes, this is pretty good, unless you get very close to the black hole.
F = G M1 M2 / R^2 ... force
F = d (M v ) / dt .... acceleration is dv / dt, so if M is constant, we have F = M a
At the most specific level, we have the non-linear Einstein field equations (10 of them): see the first link. But these are much too complicated to solve, except in a few special cases. So for practical calculation of planetary orbits, we have the parametrized post-Newtonian (PPN) equations at the second link (see equations 8-1 and 8-2). There are also equations that consider that the Sun, Earth, and Moon are not perfect spheres.
Just for fun, I include a link to one of my favorite lectures on gravity.