How can I create a Views field template per display? I successfully could create a template file for views field after reading this page.

https://api.drupal.org/api/views/theme!theme.inc/group/views_templates/6

However views-view-fields--myviewsname.tpl.php override all the displays in "myviewsname"

Is there any way I can make a template that overrides just one display in a View?

For example, a template that overrides just "display-1" in "myviewsname"
views-view-fields--myviewsname--display-1.tpl.php will do it, eg, views-view-fields--VIEWNAME--DISPLAYNAME.tpl.php.  (Along with changing any underscores in the names to single dashes)

On the api page you reference, you can see this in their example as the display mentioned there is called page.

These and other template names can be easily discerned from clicking on the Theme Information link in the Style Settings in the Views UI.