I’m creating this category because this kind of saturators are crucial contributors to to getting a clean and loud master. They can gracefully shave the peaks levels touching 0 dB Full Scale and output anywhere between -0.7 dB to -1.5 dB lower peak. That’s a whole lot of headroom in a master bus chain or mastering context.
Due to their transfer curve shape, the waveform folds back when the peaks cross over 0 dB, so they can sound nasty if they don't have a hard clipper onboard, so it goes without saying that they’re often ignored because they’re misused.
I remember people cranking up MixHead until they really hear something, hear the distortion, and then conclude that the distortion is nasty and discard the tool. These saturators don't suppose to have an obvious sound, they just have to subtly touch the peaks. If you're asking why bother with such a tool instead of a compressor or fast limiter, the answer is that the wave shaping delivers a more solid, consistent body of the sound, because it works only at the top of the offending samples. Dynamics, due to their release parameter, always affect the samples that follow the triggering sample, and the resulting sound is blurred, less consistent. We want to use waveshaping as much as possible - that is, before the mix sounds audibly distorted.
So here’s the strategy: make sure the chorus of the mix doesn't cross over 0 dB on the master bus, have a hard clipper (doing nothing, but for safety purposes) right before this kind of saturators, and then whatever processes come after (multiband compression, or a hard clipper and limiter combo).
Most of these saturators have one set transfer function, it’s only Sonnox Inflator that has a slider that can change the shape of the transfer function.
Finally, why would you chose between one or the other? The ones with just a single 3rd are the cleanest, the ones with a bith of 5th can add extra energy and still sound transparent in a busy mix. Being saturators, they’re highly level dependent, and in a mix usually the low end is the loudest. This means that the low end will trigger the saturation, whitch in turn will project higher order harmonics into the low midrange. So basically the choice is made on how the 100 Hz to 1 kHz sounds after saturation, that's why it worth having around few of them.
Be advised, the differences are subtle, but we use them primarily to gain clean headroom, not to enrich the mix with harmonics. We go for utilitarian, not for aesthetic.
On the logic above, most saturators are too rich in harmonics (most tape emulations, tube plugins like Black Box HG-2, iZotope Ozone Exciter) and they switch too sudden from doing nothing to audible saturation. On the other hand, Saturn 2 from FabFilter, has the Clean tube and tape that are too rich, and then the transformer that does almost nothing.
I do encourage anyone to do their own set of testing (sine wave and oscilloscope and FFT) including your DAW stock saturators, to have a better grasp of their richness and potential use.
Metric Halo & Make Believe MixHead
A reverse engineered (not cloned) SPL Machine Head hardware, with some tweaks (a little 5th in addition to the main 3rd.
Plugin Alliance & SPL Machine Head
Released after MixHead. The SPL hardware is a digital box (no analog inputs or outputs and no analog processing). It was a hardware made after a plugin named Magneto. The initial code was made by SpectralDesign (a young DSP coder that went to TC Electronic in their glory years as senior dev, and currently is Principal Scientist at Adobe), licensed to Steinberg and Cube-Tec. Steinberg made upgrades to the initial code and, and completely rewrote the code for Magneto II which now is part of their channel strip, but needless to say, because they added extra upper harmonics, it doesn't sound good for master bus purposes.
For whatever reason, (confirmed by their own YouTube video in German), SPL did not retrieved the DSP code from inside their own hardware, but they re-licensed the code from Steinberg, the original version that came from SpectralDesign.
The SPL Machine Head (plugin) outputs one single 3rd harmonic. The problem is that it seems to have a built-in limiter to avoid the foldback or clipping, and that makes the sound slightly soft. (The hardware is running in fixed point math and it’s hard clipping.) This is the reason I chose MixHead between these two.