Why is gravity generally defined as the consequence of a curve rather than a pull? This question is asking to better understand the semantics of mainstream physics. My assumption being there is a specific narrative behind the preferred term. In any case, it seems like an essential topic of mainstream physics worth of reflection.
Given we do not know what gravity "is" in any fundamental way, and we are defining its behavior from an external frame of reference, the space-time 'bend' analogy comprises a 'localized' stretch of space as well as a 'localized' dilation of time, but does not seem to semantically account for the gravitational 'pull' itself.
My question respectfully and genuinely asks: why does it seem more appropriate to define the unidirectional gravitational force exerted by any given mass or energy as a mere 'warp', 'bend' or 'curvature' of the space-time fabric instead of a continuous 'pull' of the three-dimensional hypersurface itself. The unidirectional nature of gravity seems more akin to a continuous 'pull' than a bidirectional 'curve'. Conversely, imagine a spaceship moving through a space-time grid at a constant acceleration of 1 g, would you describe it as 'bending' time and space, or 'pulling' through time and space?
Semantically speaking, calling it a 'pull' does not preclude its effect from aligning with a geometric 'curve' (speed per time dilation & direction per space deviation).
I'm not proposing any new theory, nor I believe this question challenges GR in any practical way (time dilation, gravitational lensing, etc). My concern being the semantics getting in the way of a more comprehensive yet intuitive understanding of it.
 A: The terminology is used for two reasons.
First, the mathematics used in general relativity for describing gravity is (pseudo) Riemannian geometry. As a result much of the terminology comes from the pre-existing terminology of Riemannian geometry.
Euclidean geometry has a series of axioms which are valid on a flat plane, but which are invalid on a curved surface like a sphere. Riemannian geometry is concerned with the geometrical concepts related to curved surfaces like the surface of a sphere where those axioms do not hold. As a result much of the terminology was developed to reflect the geometrical impact of working in an arbitrarily curved surface. General relativity borrowed the math to describe gravity and the existing terminology came with it.
Second, there are gravitational effects in general relativity that don’t align well with the force concept but do align well with the geometrical concept. For example, gravitational time dilation.
There is no a priori reason that gravitational time dilation would be predicted from a force, and yet it falls out naturally from the geometrical approach and formed one of the earliest experimental triumphs of general relativity. Because the force concept provides no insight to such gravitational effects but the curvature concept does, the terminology makes sense. Furthermore, geometry is highly intuitive, with many insights gained through reasoning about well known curved surfaces.
Finally, it is possible to discuss the force of gravity. The force of gravity is given by the Riemannian formalism as what is known as the Christoffel symbols. In that context the gravitational force has the same status as the centrifugal and Coriolis forces. As such forces make some people uncomfortable or are considered “fictitious” there is a reluctance to adopt that terminology.
A: Imagine that you are in a free-falling elevator. You will feel as if you are weightless. There is no way you could tell whether you are falling in a gravitational field, or whether you are in the depths of space, far from any source of gravity. Any experiment you can do will have the same results in both cases.
Actually, there is only one way to detect the presence of gravity -- if you could look out the window and see another elevator falling beside you, you would notice the elevator slowly coming closer to you. This is because both elevators are being drawn toward the center of the earth. In the absence of a gravitational field, any two objects that are "free-falling" (have no forces acting on them) will NOT be drawn closer to each other. In a gravitational field, two objects that are free-falling (have no forces other than gravity acting on them), may be drawn closer together. This is called a tidal effect, and it is gravity's only effect.
This may seem very counter-intuitive. On the surface of the earth, we think that we feel the force of gravity. Actually, though, the force that we feel is the force of the ground preventing us from continuing on our natural, free-falling path.
Now the reason curvature enters into the picture is that the mathematical concept of curvature allows us quantify the effects of a force that is only detectable through its tidal effects. You can imagine two ants crawling in a straight line on a flat surface. If they start out going in the same direction, their paths will never cross. Now imagine they are crawling on a curved surface, like the surface of an apple. Then even if they try their best to go in a straight line, they will find that they are sometimes drawn closer to each other because of the curvature of the apple. The modern view of gravity is that all objects attempt to travel in straight lines, but that gravity bends space. Because of this, objects travelling in "straight" lines may be drawn closer to each other as a result.
Edit
You might object to the above reasoning by saying that the falling motion of the elevator is simply offsetting the gravitational effect. That is certainly one way of looking at it. However, since gravity effects all particles (unlike any other force), there is no way to tell whether the elevator is in a gravitational field or not except to look at tidal effects. It is actually conceptually (and mathematically) much simpler to take the view that all particles naturally attempt to move in straight lines, and the curvature caused by gravity can cause them to come together. This approach has led to many useful new predictions that have been experimentally confirmed. Technically, though, you could view gravity as some kind of force if you want. It's just much harder to account for all the new effects when you do it that way.
Edit 2
It can be hard to reconcile this geometrical point of view with everyday experience. As an example, consider two massive balls, at rest with respect to each other in empty space. It can be hard to understand why curvature would cause these two balls to come together, since they are not moving through space. The key thing, however, is that they ARE moving through time. In relativity, space and time are not two separate concepts. Gravity curves both space and time, so as the balls move through time, the gravitational curvature will cause them to bend into each other.
A: I think that the idea of gravity as a curvature of the spacetime leads to the notion that even the $mg$ force we feel comes from it, what is misleading.
For example: all that class of high school problems where it is said to assume $g$ constant and ignore the air resistance are using an approximation of flat spacetime. Because the situation is totally similar to be uniformly accelerated.
That means: half of our idea of what is gravity, either intuitive, either learned in the school is not related to spacetime curvature.
The other half, related to the orbit of planets or satelites requires spacetime curvature. But that difference is normally not mentioned.
