You're missing everything, because language is misleading. There are subconscious meanings to words that color your reasoning and lead you off into the ozone. Stick with the definitions.

A moving object has kinetic energy that depends on how much velocity it has. It changes speed by accelerating (acceleration means "change of velocity", nothing else). In fact, sometimes it is abbreviated as delta-V, Capital Greek Delta is shorthand for 'change'. It doesn't matter how it accelerated, how long it took or where or when it happened, or how many times you accelerated to that speed, it just matters how fast it is going. In other words, the acceleration is not important, it is just the velocity that is important.

Your sentence about the field around a hammer makes no physical sense. I suspect you visualize a hammer as being surrounded by a cloud of something. It isn't. In physics, just because a sentence is grammatically correct doesn't mean it has anything to do with the real world.

Your making all this more complicated than it really is, and you are inventing things that you feel are required in order to explain that complexity.

A mass stationary to its surroundings has no kinetic energy.

If the mass moves, it is experiencing a force, whether it be from the curvature of space time (gravity) or a hammer.

The change in velocity of that mass is its acceleration, its mass times the acceleration is the force on it.
(F = ma). In the case of the hammer, the force and acceleration occur very quickly.
If the mass is falling, the acceleration and force are constant and gradual.

When the acceleration stops the mass continues at a constant velocity, v = at = change in position with respect to time.

Once the mass is traveling at that velocity it has kinetic energy E = 1/2mv**2.

All of these numbers and letters have to do with the mass and its motion, they do not come from any field. The motion comes from a force, which implies contact with another mass or a gravitational force.

Consider a stationary ball on a billiard table. It is approached by another identical ball moving at some velocity v. When the second ball hits the first, its kinetic energy is now available to both balls to share. One or both of them will now be moving so that their combined kinetic energy is the same as the original kinetic energy that existed before the collision.

You don't know enough physics yet to predict what will happen, (because you haven't learned about momentum, another little bookkeeping tool physicists use) so I'll tell you.

The first ball accelerates to a full stop. The second ball accelerates into motion.
That is, the second ball exerts a force on the first, causing that mass to accelerate. The second ball exerts an equal and opposite force to the first, causing it to stop. F = ma. And forces always occur in equal and opposite pairs according to Newton.

The second ball is now moving at the same speed as the first one was, so it has the same energy. In fact, the total energy available before and after the collision is conserved (we ignore friction with the air, the cloth surface of the table, etc)

Now how do we know that the second ball stops cold and the first ball takes off with its speed? Because I spent too much of my youth playing billiards, and because of the law of conservation of momentum. Experience (that is, observation and experiment)has taught us than in all interactions, energy and momentum are conserved, they do not change before and after there is an exchange of forces. In order for energy and momentum to be conserved we have defined these quantities so that they are. Conservation of momentum is what tells you how the energy involved in that collision wull be shared between the two balls. Conservation of energy tells you how much is available to be shared.

The last time I explained this to you (you obviously were not listening) I used the example of an astronaut firing a pistol in weightless space, and by knowing the mass and velocity of both the astronaut and the bullet, it was possible to calculate how fast the astronaut would recoil. Since both energy (1/2mv**2) and momentum (mv) are conserved, you know that the potential energy available in the cartridge gunpowder will be divided between the man and the bullet (but not equally, since they have different masses. The momentum is also conserved, and it is zero, at the beginning of the experiment, so it must be zero afterwards.

The momentum of the bullet is mv.
The momentum of the man is MV (we'll use uppercase for the man). They add up to zero, so mv = MV.
Rearranging, m/M = V/v. The ratio of velocities is the inverse of the mass ratio, so if the man, spacesuit and pistol weigh, say, 500 times as much as the bullet, they will recoil at a velocity 1/500 as fast. (I ignore what happens to the gases of the gunpowder combustion, etc.)

The point I'm making is that with the laws of conservation of energy and momentum you can always determine how masses and velocities are shared (through exchange of forces) in collisions, rockets, orbits, explosions, pendulums, rotating bodies, springs, bouncing balls, electrons in magnetic fields--whatever.

You don't need to know "where it comes from". All science cares is what happens, not why. That way lies madness.