I am working on a error state Kalman filter to estimate the position and orientation of a flying vehicle using IMU data for state propagation and vision based measurements for updating.
So for what I know/understood from "Quaternion kinematics for the error-state KF": I have the nominal state $\mathbf{x}$ and the error state $\delta \mathbf{x}$. $\mathbf{x}$ is propagated using my IMU measurements. The $\mathbf{F}$ and $\mathbf{H}$ matrices are defined based on the error state and not the full state. The error state is not propagated.
$\delta \mathbf{x}$ is computed in the updated step for the first time, using:
$\delta \mathbf{x} = \mathbf{K}\left(\mathbf{y} - \mathbf{H}(\hat{x})\right)$ --> the error state is independent of any previous estimates of the error state
For what I understood this is the first time $\delta \mathbf{x}$ is computed, and after it is used to inject the error state in the normal state it is re-set to zero.
So for what I understood we do not need to predict/propagate the error state as we have no information on the errors and it is thus initiated with zeros, the error state will always return zero. The covariance matrix obviously has to be updated as the error grows.
Is it correct that the error state is not predicted in the prediction step and only used during the update step?
Yes - that's exactly right. After you've processed measurements and computed delta-x the error values are incorporated back into the whole state x. Then delta-x is zeroed-out before being used for the next set of measurements.