According to Analytical Mechanics of Space Systems, Osculating Elements are the one containing different perturbations in the Orbit. When you omit the short time perturbation (J2) you gain the mean orbital element. The mean orbital elements change not much during the short time so they are a good choice for long term orbital maneuvers (when you have small thruster and are interested to see the changes in orbit in a long term basis).
The whole concept of osculating to mean came with two paper from 1959:
- The motion of a close earth satellite Kozai
- Solution of the problem of artificial satellite theory without drag Brouwer
They did not cover all the orbits, so Lyddane done some correction to them:
- Numerical comparison between Brouwer's theory and solution by Cowell's method for the orbit of an artificial satellite Lyddane
These are relevant for Keplerian Elements.
Books and Papers which can help you for Equinoctial Elements:
- Analytical Mechanics of Space Systems Schaub and Jenkins
- An Introduction to Astrodynamics Battin
- On the Equinoctial Orbit Elements Burke and Cefola
- The Long-Term prediction of artificial Satellite Orbit Cefola
- A computerized algebraic utility for the construction of nonsingular satellite theories Zeis
If you want to test your implementation or the mathematics use Orekit, it has most of these orbit types implemented.