Classification of Forms and understanding what goes into \$PrintForms, \$OutputForms, Out[x]//XXXForm=, and what does not

Question

Mathematica has lots of Forms. Some of those appear in $PrintForms, like InputForm, OutputForm, CForm and so on, but some Forms do not, like ScientificForm, and StringForm or HornerForm.

What distinguishes when something goes into $PrintForms and what doesn't?

Edit: As noted in the comment, https://mathematica.stackexchange.com/a/73539/1871 mentions there is also $OutputForms which seems to include more things than $Printforms. In particular it contains ScientificForm, but not StringForm or HornerForm.

What is the difference between these $PrintForms and $OutputForms?

In https://reference.wolfram.com/language/tutorial/TextualInputAndOutput.html#12368 I read:

It is important to understand that in a typical Wolfram Language session In[n] and Out[n] record only the underlying expressions that are processed, not the textual representations that happen to be used for their input or output.

If you explicitly request a particular kind of output, say by using TraditionalForm[expr], then what you get will be labeled with Out[n]//TraditionalForm. This indicates that what you are seeing is expr//TraditionalForm, even though the value of Out[n] itself is just expr.

I note that StringForm and HornerForm do not work that way though. Is there guide to understanding what will be marked Out[n]//XXXForm and what won't?

More generally, are there natural classes of forms? Two kinds suggested above are: those that are in $PrintForms, $OutputForms, those that appear as //XXForm in output, and others.

However these might be quirky and there might be other more natural classifications of Form functions.