Custom taxonomy templates

Hello great people of wp community.
I got a weird problem (or I just don’t get it).

I created a custom post type and a custom taxonomy like this.

As the title says problem is with their “category” (tax) pages.
I have these files with only header,file name and footer.

• archive-cat.php
• archive-post.php
• archive-product.php
• archive.php
• category.php
• home.php
• taxonomy-cat.php
• taxonomy-product.php
• taxonomy.php

These all pages are just for testing purposes. All I would like is to split “category” (tax) page and blog archive still even with all these files both:

• localhost/cat/cat-term and
• localhost/news (selected blog page)

loads home.php, if I remove home.php they both load index.php.
I tried setting permalink in CMS to plain and then back to post name as this solved most of similar problems on these forums but no luck.

Anyone have an idea what could I do next?
Thank you for your time.