What is the difference between these two observational update models in an information filter? The Information Filter is defined as the mathematical inverse of the Kalman filter. As defined in this Wikipedia article, the observation update of the Information Matrix is defined as $$y_{k|k} = y_{k|k-1} + H^T_kR^{-1}_kz_k$$
This definition holds consistent in the article's source reference here (pp 9, eq 55) and that source's reference here (pp 264, table 6.20).
Looking at some additional literature, this measurement function is sometimes defined differently. This source on the Extended Information Filter (pp 698, eqn 18 presents a definition (albeit in alternative symbols) that reads, once reordered and using the above standard symbol set, $$y_{k|k} = y_{k|k-1} + H^T_kR^{-1}_k(z_k - h(\bar X_{k|k-1}) + H_k\bar X_{k|k-1})$$
That is, the measurement is now a quantity of the measurement minus a non-linear prediction of the measurement from the state estimate plus a linear prediction of the measurement. The cited paper seems to describes this as a Gaussian approximation of a non-linear measurement.
What I'd like to know is:
What is the second form of the equation implying geometrically?
When is the appropriate time to use this second form?

I have a partial answer to my own question.
$(z_k - h(\bar X_{k|k-1}) + H_k\bar X_{k|k-1})$ is the result of a Taylor Series Expansion which aims to linearize the residual calculation.
Edit:
I found this set of lecture notes (section 1.3) that dives into Taylor Series Expansion in the application of the Extended Kalman Filter. Essentially it strives to approximate the non-linear measurement model. Sections 1.5 also has some great commentary on applications and vulnerabilities.
This really helped me dig down into the performance of my residual calculation ($y = z - Hx$) and to look at what the components of the Tayler Series Expansion is supposed to do: add in an error approximation component to "make up for" the error in the "predicted measurement" $Hx$.
Ultimately, it seems my system was too nonlinear to get good approximations even with this technique. Examining the role of the risidual - to "steer" the Kalman gain towards the measurement or the model - helped me identify inductively how to structure the residual calculation.
Going to mark this as answered.
