Different styling for each teaser display How to style Teasers differently in different displays and pages? 

I am struggling very much with Drupal's CSS classes for teasers....Is there anyway to add them separate class name for each case? 

Now, if I'm styling Teasers on Taxonomy page - the same class applies to Teasers on Front Page Promoted teasers, in Teaser blocks in Views, etc....

Well, I removed "promoted" teasers and put them in Views block and assigned a class; and it's separate on Taxonomy page....

BUT - what would I do if I would wish to have a separate template on some other (taxonomy) page for...teasers? 

All those Teasers are in one big pot, named .node-teaser and the same field names...

//Using D7 and Zen 7.x.5x.
You can create different Entity View modes instead of the single teaser. While you can do that via a custom module in Drupal 7, there is a nice little module for that (https://www.drupal.org/project/entity_view_mode) too.

Create different modes like 'teaser_front', 'teaser_sidebar' and theme them differently using the standard theming convention (can also be found in the module page).