Natural motion of objects according to Einstein

In addition to what Jacopo says, I would elaborate on the part where you say "the geodesic path will change". Change relative to what? Newton assumed the presence of Euclidean space through which objects move, so we could say the path of a Newtonian object changes, ie. curves, relative to space itself. But in relativity, that Euclidean background space is gone.

Instead, what we can say is that different geodesics "change" relative to each other. Maybe the most important example is that two geodesics can converge or diverge - that's the concept embodied by the Ricci tensor which you'll find in the Einstein field equation. Think of a sphere, where geodesics are great circles - two geodesics leave the north pole moving away from each other, but they magically re-converge at the south pole. Of course that's because the sphere is curved. Convergence corresponds to positive curvature, while divergence is negative curvature (like a potato chip or a saddle).

In spacetime, the type of geodesic convergence that's most obvious to us is convergence along the time direction: when you throw a ball up and it comes back down, it's "spacetime direction" (really its velocity) is initially at an angle relative to the Earth's (think of a graph of the ball's height vs. time), but the two paths meet again when the ball hits the ground - so spacetime must be positively curved along the Earth's time direction. In fact the equation says it's curved along spatial directions as well, we just don't notice.

...and lead to the illusion of a gravitational force.

I prefer to say that GR chooses a metric, so that the velocity is constant. But that velocity is the covariant derivative of the coordinates, and includes the derivation of the time coordinate (it is the 4-velocity).

The idea is better understood in 2 spatial dimensions instead of 3 spatial and 1 time dimensions of GR.

Flying from Tokyo to Paris, the airplane follows the shortest path as possible (except for procedure guidelines or avoiding unsafe countries). If we use a string in a globe we find very close the actual path.

But if we check a compass time to time, it becomes clear that the route is not constant. In the first hours the plane has a North component, and in the last hours a South component. Most of the trip is over Russia, at latitudes higher than Tokyo and Paris. So it seems that the movement is accelerated, because the direction of the velocity is variable.

The metric here is our usual latitudes and longitudes coordinates. The covariant derivative is the mathematical tool that obtains the shortest path (in this case a great circle) between the 2 cities, correcting the simple derivative, that changes direction all the time during the fly.

In the case of gravity, our perception shows us accelerated movements and the existence of a force. GR models a mathematical frame where there is no force and no acceleration.

It is a way to correct perceptions, like when we say that the earth is rotating and the daily movement of sun and stars is apparent.

if the differences between the view of Einstein vs. Newton in regards to the natural motion of objects could be described in a better way or be made more precise

Newtonian mechanics can also be described in such a way that freely falling objects follow geodesics in a curved spacetime. This "Newton-Cartan" spacetime has an affine connection and a curvature tensor, but is not Riemannian or pseudo-Riemannian (it does not have a nondegenerate metric). "Geodesic" here means "straight" (following the affine connection) but not "shortest".