How do I use a term page display?
Both of the modules mentioned (Taxonomy display and TVI) that make Taxonomy term pages easier to configure rely on Views, so you do need to have the Views sorted out before you can assign different Views to different Vocabularies. These modules don't replace the "Views magic". 

If you don't use these modules that override how term pages are displayed the "Taxonomy term" Views that comes out-of-box handles all of the terms equally. BTW "Taxonomy term" Views is of the content type (not the taxonomy term type, the name can be confusing a bit). 

But the "Taxonomy term" Views can be changed to handle different Vocabularies differently. This change can be made in the Contextual filter in the validation settings. Then you can clone that page, give it a different path, and adjust the contextual filter validation to a different Vocabulary. 

Both of your cases can be handled with the modified "Taxonomy term" views. Using the "Summary" option in the contextual filter for the Taxonomy reference field will list the terms present in the views results. When you click through those links you see the content listed. Although your "No results behavior" solution also works. 

You can get the Parent info into your path through another Contextual filter (Taxonomy Name), that you can get after adding the Relationship for the parent term in your Views. 

It's all in the Contextual filters and perhaps some Relationships to create your cases. There are a lot of answers here relating to those so you can search and learn from them. 

It is also mentioned in the comments you can use the theme templates to override how Term pages are displayed, when you're not using the Views to build those pages, but you'll probably still use Views to build blocks to put in those templates, unless you know SQL and can create better queries manually.