Two things are happening here.
The first is the replacement of the actual latitude phi_n by the "conformal latitude" phi_nc. Think of this as distorting the ellipsoid (as specified by r_eq and e) into a perfect sphere. Because it is an ellipsoid of revolution, no change to the longitude occurs (lambda_n = lambda_nc), but the latitudes shift slightly. They do so in a way that is locally angle-preserving ("conformal").
The second is a slight adjustment of the scale of the stereographic projection, also to account for the shape of the ellipsoid. This is reflected in the variable c, which you can see depends on the eccentricity e (the sole determiner of the ellipsoid's shape).
Here is John Snyder's account:
...the ellipsoidal forms of the Stereographic projection are nonperspective, in order to preserve conformality. The oblique and equatorial aspects are also slightly nonazimuthal for the same reason. The formulas result from replacing geodetic latitude phi in the spherical equations with conformal latitude chi ... followed by a small adjustment to the scale at the center of projection.
Map Projections--A Working Manual, p. 160, emphasis added.
Incidentally, "null distortion coordinate" is an idiosyncratic term. According to Google, this thread is the only place on the Internet where such a phrase occurs!
Personally, for the cliff's notes I find the ESRI ArcGIS desktop software help system useful, and and also the ESRI book Understanding Map Projections, its first 30 pages are not unlike a short textbook, followed by ~70 pages of appendix on individual projections, their uses, strengths, weaknesses, etc.
From these, you'll quickly understand all the pieces of a complete SR - spheroid + datum + meridian + unit of measure => GCS; and projection + parameters + unit of measure + GCS => PCS.
For all the math, try the classic USGS Professional Paper 1395 by John P. Synder - Map Projections - A Working Manual. You might find a PDF somewhere if you google it.
Finally, take a look at the 'documentation' links at the bottom of the page for the Proj.4 library.
And as a DB of spatial references, I tend to refer to http://www.spatialreference.org/ or another ESRI list provided as part of the docs for their ArcGIS Server REST API.
Hope that helps.
Best Answer
Try the EPSG Geodetic Parameter Dataset which is more up-to-date and where epsg.io got its information.
Admittedly, it doesn't always have projected coordinate reference systems that are suitable for an entire country--but only because no one's told us what is used for a map of the entire country!
Spatial Reference allows user-contributions so might have more information.
Disclosure: I'm on the subcommittee that maintains it.