Perhaps I can be of some assistance here.
Gravity (and its acceleration) specifically refers to a force which all massive objects are theorized to exert on each other to cause gravitation.
You are correct in that gravity, when it refers to a force, does refer to this force. There are some theories of physics which include this force; there are others that do not. One that does is Newtonian mechanics; one that does not is General Relativity, and since General Relativity is more accurate on a wider range of scales than Newtonian Mechanics, I will assume it to be true for the remainder of this post.
Thus, gravity is not a force of any kind. Instead, it is a property that nearby straight lines tend to get closer to one another over time; in GR this is formally called "convergence of nearby geodesics".
The acceleration of an object (for example a car on the surface of the earth) is the rate of change of its velocity.
This is almost correct; you did not specify the frame of reference in which the car's (or other object's... but I'll stick with the car example) velocity is being measured. Sure, you could put lines on the road and measure how quickly the lines go by; or you could measure the circumference of the wheels and measure how fast they are turning. These would be measuring velocity with respect to the road.
However, you could also measure the car's velocity by bouncing a laser off of the moon. You would get a different number for your velocity now, since you are using a different reference frame. You would also get a different number for acceleration (you might be moving at a constant velocity w.r.t. the road, but a variable velocity w.r.t. the moon, which is moving in a circle). You could again measure velocity w.r.t. the fixed stars by assuming a certain shape of the Earth and looking at the visible sky; this would be different from both the moon-measurements and the road measurements. You could even measure velocity by looking at how quickly the cars on the road are moving past you... for each car, you would get a different measurement. In particular, you could measure your velocity and your acceleration relative to yourself, and you would get zero!
So, which is correct? Well, the whole point of theories with the word "relativity" in the name is that it is not meaningful to ask which is correct; they are all correct
in the reference frame in which they are measured. Special Relativity is the promise that you are allowed to measure your velocity with respect to any reference frame you want (i.e. that velocity is relative) -- in particular, you get to declare yourself to be at rest. It gets around the weird side effects by assuming that time and distance are not defined globally.
General Relativity is the promise that you are allowed to measure your
acceleration in any reference frame you want (i.e. that acceleration is relative) -- in particular, you get to declare yourself to be not accelerating.
How does General Relativity make this promise? Well, it says, spacetime is curved; objects moving in straight lines might appear to be moving in curved lines from the perspective of other objects. Thus, if you think you are accelerating but want to claim that you are not, you can instead just blame the apparent change in velocity on the curvature of spacetime.
To restate: you are permitted to believe either that you are accelerating,
or that spacetime is curved in such a way as to make it look like you are. It doesn't matter what "force" is causing you to accelerate: your car's engine, a rocket, a magnet, an elevator cable, whatever; doesn't matter. No acceleration is happening
to you; it's only happening to other things, and it's happening, (you are allowed to believe, and be correct) because spacetime is curved. Since "spacetime is curved" is what GR means by "gravity", then GR is saying that from your perspective, every acceleration is gravity.