Q: What does an accelerometer actually measure?

A: Normal force per unit mass.

Note that it’s not the net force per unit mass (which would be acceleration), but it is the normal force per unit mass. This unusual quantity corresponds with what a rider on a roller coaster feels during the turns. This interpretation is useful even for the scalar total acceleration value, which is 9.8 N/kg for a three-axis accelerometer at rest, zero for one in free fall, and greater than 9.8 N/kg for one making a corner. This normal force interpretation works even for a one-axis accelerometer being accelerated in a horizontal direction. The reading is non-zero as the test mass inside the device has to have a force applied to accelerate it. That’s just a normal force that happens to be horizontal.
When discussing the accelerometer reading, we can call it the Normal Force per Unit Mass, with units of N/kg.

Q: I thought the Accelerometer measured acceleration!

A: Here we are being very careful to not call something an acceleration when it is not a kinematic acceleration. For example, an “acceleration” of 9.8 m/s2 for an object that remains at rest is clearly a problematic interpretation, yet that’s what the accelerometer reads.

You can correct the Accelerometer reading to get a true acceleration by adding the component of the gravitational acceleration field along the direction of the sensor arrow. For example, if the axis of the accelerometer is pointing upward, then the gravitational component is -9.8 m/s2. The Accelerometer reads 9.8 m/s2 when the arrow is upward and the device is at rest. By adding -9.8 m/s2, we get zero, which is the correct acceleration. If the arrow is horizontal, then the reading is zero, but the gravitational component is zero, and we still have zero for the true acceleration. If an Accelerometer is zeroed to remove the influence of gravity, the zeroing is later wrong if the Accelerometer is rotated.