Jan, The sloshing model is marginally different than the ENSO models that are of current interest and considered the standard models for ENSO. There is the Zebiac-Kane model and various delayed action oscillators, which are described on the Azimuth Project http://www.azimuthproject.org/azimuth/show/ENSO
But those IMO are complicated by the fact that they add nonlinear terms -- I think on the grounds that someone wanted to see chaotic behavior. What happens if you remove the nonlinear terms is that you end up with the basic 2nd-order differential equation called the wave equation, which is the one I have been using recently. The Mathieu sloshing equation is a slight perturbation on the wave equation, which verges toward chaotic behavior, but is a long way from the chaos of the Zebiac-Kane type models. I started with the Mathieu (see note ** below), and frankly probably wasted a of time because I thought that would be my entry in weaning off a fully chaotic model.
Allan Clarke at FSU is the biggest proponent of wave equation type models for ENSO. Read this paper  and you can see how the derivation of the wave equation occurs. It is spelled out in its full uncomplicated glory
This formulation very hard to argue against because it is a simple model. But if you ever see anyone use it in the literature with periodic forcing terms applied, I would like to know about it.
Thinking about the larger-scale ocean patterns is not on my radar as I am focused on this single standing wave mode.
 A. J. Clarke, S. Van Gorder, and G. Colantuono, “Wind stress curl and ENSO discharge/recharge in the equatorial Pacific,” Journal of physical oceanography, vol. 37, no. 4, pp. 1077–1091, 2007.
** I remember reading an aside in one article where the authors dismissed a Mathieu equation type of formulation because the geometry of a spherical earth does not allow the boundary conditions necessary for that kind of solution.