Display current taxonomy slug in a post

There isn’t a concept of a post mapping to a single taxonomy, so this won’t work out.

Taxonomies are things like colour and size with terms such as lilac, azure, maroon, small, medium and large.

So your question is a bit like asking to be able to display “colour” when on a post, whereas the real mappings may well be to blue for colour and medium for size.

Are you able to put some real names into your example? If we could see what data you are working with then we might be able to suggest a structure that would work for you.

Edit: Expanded answer based on clearer information from OP.

You’re misunderstanding slightly how taxonomies, terms and posts are structured. The taxonomy holds a set of terms which are like labels to apply to your posts. Your posts don’t sit within taxonomies and terms like files within folders, rather the terms tag the posts. Terms within a taxonomy can be hierarchical and I think this is what you need.

Create one taxonomy called something like “type of activity.” Make it hierarchical so that you can then add nested terms like this:

 - running (Term)
 - watersports (Term)
   - swimming (Term)
   - diving (Term)
   - boating (Term)
   - Kiteboating (Term)
 - materials arts (Term)
 - teamsports (Term)

Then instead of trying display the name of a taxonomy, you want to display the name of the top level term. You can get a list of just the top level terms using:

$args = array(
    'taxonomy' => 'activity_type',
    'parent' => 0,
    'hide_empty' => false           
);
$terms = get_terms( $args );

You can get all the top level terms that apply to the current post like this:

$args = array(
    'parent' => 0,          
);  
$terms = wp_get_object_terms( $postID, 'activity_type', $args );