I think it’s worth clarifying a bit of terminology first:
- Taxonomy – a set of related terms
- Term – an entry (I want to just say “term”) in a taxonomy
- Categories – a taxonomy that’s in the default WordPress
- Tags – a taxonomy that’s in the default WordPress
- Category – a term in the Categories taxonomy
- Tag – a term in the Tags taxonomy
- Custom taxonomy – a non-default taxonomy in WordPress
So the taxonomy-{taxonomy-name}.php template targets all terms in a specific taxonomy. taxonomy-{taxonomy-name}-{term}.php targets a specific term in a taxonomy.
I’d encourage you to take a look at the Template Hierarchy flow chart from the Codex. From what you describe it sounds like you either need to use archive.php that affects all archives (assuming you haven’t specified a more-specific template) or taxonomy.php which targets all Custom Taxonomy archives (so not Categories or Tags).
