What does the balance ball actually indicate?
1. Preamble

The ball reacts to the body force (or acceleration), with the exact same working principle as a bubble level. In airplane frame, the force is expressed as:

$$Y+mg\cos{\theta}\sin{\phi}=m(\dot{v}+ru-pw)$$

where $Y$ is all external side forces minus gravitational component, $m$ is airplane mass, $g$ is gravitational acceleration, $\theta$ is pitch angle, $\phi$ is roll angle, $\begin{bmatrix}u &amp; v &amp; w\end{bmatrix}^T$ is the inertial velocity in vehicle frame, $\begin{bmatrix}p &amp; q &amp; r\end{bmatrix}^T$ is the angular velocity in vehicle frame.

First, as any accelerometer, the ball does not react to gravity. Second, and this is a preference in bookkeeping, you can consider the accelerometer (or the ball) to only react to the external forces (i.e. $Y$), or only react to the inertial forces minus gravity (i.e. the right side of the equation minus the gravitational component). 

2. In Flight

Taking thrust out of the question, then in flight, the only thing that can contribute to $Y$ is aerodynamic forces. What's the largest source of lateral force in air? Sideslip. There you have the relation between ball measurement and sideslip angle measurement. If you're unconvinced, read the sub-sections below.

2.1 Symmetric Thrust and Airframe

For symmetric thrust and airframe, the only yaw moment sources come from aerodynamics, and the largest contributors are sideslip and rudder (ailerons and spoilers also contribute due to differential drag, but the effects are even smaller).

Using the B737 model in AVL at Mach 0, flap 0 and AOA 0 for a rough idea, -5 deg rudder generates about 4.4deg of sideslip (nose left), at which point their yaw moments cancel each other out. However, the total $C_y$ is 0.055 to the right.

Comparing the stability/control derivatives (per rad), $C_{y_\beta}$ is -1.2, whereas $C_{y_{\delta_r}}$ is 0.44, which means that at the aforementioned conditions, the sideslip is overpowering the rudder by 140%, in terms of side force.

2.2 Asymmetric Thrust

In the case of one engine inoperative (OEI), we need to overcome the yaw moment from thrust asymmetry via rudder.

For wings level, there is side force from rudder that must be compensated. This can be accomplished by generating more sideslip into the live engine (which means more rudder deflection is needed than to overcome the thrust yaw moment). At steady-state, the net side force is zero, the ball is centered, but there is steady non-zero sideslip.

If we allow the wings to bank toward the live engine, a portion of gravity can be used to compensate for the side force from rudder, which means that less sideslip induced side force is required to maintain steady-state (therefore less rudder required). However, since the total side force minus gravity ($Y$) is non-zero, the ball is skewed to the live engine.

In either case, with thrust asymmetry, the relationship between sideslip and side force are "broken".

3. On Ground

On ground, any point of contact with the ground provides additional sources to $Y$. This is why tilting the bubble level with your hand makes it not measure level: it's not that the level is responding to gravity, but rather that it's responding to the normal forces exacted by your hand.

For an airplane, the source is obviously coming from the landing gear contact. At high speed taxi on a straight runway, with crosswind, the sideslip creates aerodynamic side force. The landing gear resists lateral motion via an opposite side force (however, you'd need to deflect nose wheel or rudder to resist yaw moment). At steady-state, the yaw string would measure sideslip, but the net side force is zero, so the ball would measure nothing.

Things are different when it's a circular runway. Because the airplane needs to track a curvature, the combined lateral forces must provide the centripetal forces necessary to maintain the track curvature. Therefore, the ball would react to the equivalent centrifugal force.

4. Sensor Positioning

Thus far, we've been assuming that the ball is placed exactly at the airplane CG. If the sensor is offset, then it will also react to the angular rates. That's why large airplanes need to calibrate the sensor measurements to the assumed CG position. 

To answer your last question, unless the airplane is tracking linear motion, placing the sensor at different locations will give you different readings.