Set header for default posts page?

You could use a template. You just have to copy the file header.php and call the new file with the slug of the page testimonials. It would require 2 new files :

header-testimonials.php and front-page.php.

then you can call your custom header with get_header('testimonials');

See the template hierarchy. Just be sure Testimonials is your front page.

EDIT: Regarding front page (1rst page of the site) WP looks into your theme and the first file to be called is front-page.php. Most of the time WP blogs set front page as posts page (reading settings).

Here your can set another file as custom header and call it in front-page.php it will be called on front page automatically.

You could also insert some conditional tags in the main header.php but it’s better to use front-page.php when cutomizing front page.