List child terms according to parent term page
In Drupal 7, I accomplished this by creating a view (for a block) that uses the contextual filter, "Taxonomy term: Parent term" - Under "When the Filter is Not Available" choose "Provide default value", then select "Taxonomy term ID from URL", and check "Load default filter from term page". I then displayed the block on the taxonomy pages where I want the child term lists to appear.

I'm just starting to work with Drupal and I suspect that there's a more robust solution - this works with pathauto but has trouble with terms that use a URL alias - but the instructions I've found for other approaches either don't work or require considerable coding. If you devise a better way, or somebody else has ideas, I'm all ears.