The Fowler-Nordheim-type (FN-type) equation developed by Murphy and Good [Phys. Rev. 102, 1464 (1956)] contains in its exponent a correction function v, best called the "principal Schottky-Nordheim (SN) barrier function." In the last 50 years a large number of different approximations have been developed for v. This article reformulates these approximations to be functions of the scaled barrier field f rather than the Nordheim parameter y, and then compares them with the approximations v(f)approximate to 1-f+(1/6)f ln f and v(f)approximate to 1-f+0.1715f ln f recently reported. It is confirmed, as expected, that these new formulas outperform older approximations of equivalent complexity when the comparison is made over the range 0 < f < 1. However, particularly with the first formula, some older approximations are superior over limited ranges of f because they have been designed to perform well over these limited ranges. The article also presents an updated and fuller account of the theory of FN-plot analysis and an updated tabulation of values of the SN-barrier functions (also called field emission elliptic functions). Both updates use f (rather than y) as the main auxiliary variable. The limitations of the existing FN-plot theory are clarified; this reinforces the case for the development of higher quality measurements of current-voltage characteristics and of new methods of data analysis that may be able to supersede the use of FN plots.