Vielen Dank, @jhtjards! Das hat geholfen. aspect-ratio kannte ich bisher noch gar nicht!

Thanks for your support. But unfortunately it doesn’t work.
Problem still exists.

Thanks. Unfortunately it still doesn’t work.
Disabling “Check Referrer” only works with Safari but not with Chrome or Firefox.

Thanks a lot. That was helpful!

Thanks, Takis and yes you are right! The WordPress is in a subdirectory as you already discussed – I think – 2 years ago in an older thread with another person. I can’t find the thread again (unfortunately the search function of this forum is ugly).

So that you can reproduce it …

The settings in WordPress admin are:

The website:
https://domain.example/

The WordPress:
https://domain.example/site/

Then just copy the index.php and .htaccess from /site/ into root directory.

And edit the index.php to

require( dirname( __FILE__ ) . '/site/wp-blog-header.php' );

Afterwards everything works from the root-domain, but the WordPress stays in the subdirectory.

So now the problem is your plugin seems to create the link to adaptive-images-script.php in the .htaccess double. Maybe it is because the plugin is from .htaccess one more subdirectory away as well. And so it linked back to root twice.

I have created a WordPress test environment where the problem appears by default and will send you test-admin access to check it out.

Thanks a lot for your quick and great support! Your plugin is fantastic!

Thank you so much for your support!

Debug info is:

✔ PHP GD library is installed.
✖ Image cache directory has not been created.
But this is probably because the cache has not been accessed yet. 
After accessing your website from a mobile device the directory should be automatically created.

/home/www/[domainname]/site/wp-content/cache => drwxrwxrwx
✔ Installation .htaccess file is setup OK.
/home/www/[domainname]/.htaccess => -rw-r--r--
❖ Adaptive images settings dump:
array(14) {
  ["resolutions"]=>
  array(4) {
    [0]=>
    int(1920)
    [1]=>
    int(1024)
    [2]=>
    int(640)
    [3]=>
    int(480)
  }
  ["landscape"]=>
  bool(true)
  ["hidpi"]=>
  bool(true)
  ["cdn-support"]=>
  bool(true)
  ["cache-directory"]=>
  string(21) "cache/adaptive-images"
  ["watched-directories"]=>
  array(2) {
    [0]=>
    string(18) "wp-content/uploads"
    [1]=>
    string(17) "wp-content/themes"
  }
  ["jpeg-quality"]=>
  int(75)
  ["sharpen-images"]=>
  bool(true)
  ["watch-cache"]=>
  bool(true)
  ["browser-cache"]=>
  float(0)
  ["version"]=>
  string(6) "0.6.67"
  ["sanitized"]=>
  bool(true)
  ["wp-content-dir"]=>
  string(43) "/home/www/[domainname]/site/wp-content"
  ["wp-content-url"]=>
  string(45) "https://[domainname]/site/wp-content"
}

And diagnostics are:

Web Server	
Apache
Document Root	
/home/www/[domainname]/
PHP	
7.3.7-1+0~20190710.40+debian9~1.gbp032aec
PHP Time Limit	
30
PHP Memory Limit	
512M
PHP Post Max Size	
50M
PHP Upload Max Size	
50M
PHP Max Input Vars	
1000
PHP Display Errors	
No
PHP Error Log	
MySQL	
5.5.5-10.2.25-MariaDB-10.2.25+maria~stretch-log
MySQL Ext/mysqli	
Yes
MySQL Table Prefix	
wp_
MySQL DB Charset	
utf8mb4
WP	
5.2.2
WP Multisite	
No
WP Debug Mode	
No
WP Site url	
https://www.[domainname]/site
WP WP Home url	
https://www.[domainname]
WP Permalinks	
/%postname%-%category%/
WP home path	
/home/www/[domainname]/
WP content dir	
/home/www/[domainname]/site/wp-content
WP plugin dir	
/home/www/[domainname]/site/wp-content/plugins
WP content url	
https://www.[domainname]/site/wp-content
WP plugin url	
https://www.[domainname]/site/wp-content/plugins
WP Locale	
de_DE
WP Memory Limit	
512M
WP Max Upload Size	
50mb
WP Active plugins	
Adaptive Images for WordPress v.0.6.67 by Nevma
Admin Menu Editor v.1.9.1 by Janis Elsts
Anti-Spambot Email Shortcode v.1.1.5 by Robert Peake
BackWPup v.3.6.10 by Inpsyde GmbH
ChimpMate - WordPress MailChimp Assistant v.1.3.2 by Voltroid
Contact Form 7 v.5.1.4 by Takayuki Miyoshi
Duplicate Page v.3.9 by mndpsingh287
Envato Market v.2.0.1 by Envato
Favicon by RealFaviconGenerator v.1.3.15 by Philippe Bernard
Flow-Flow Lite v.3.1.23 by Looks Awesome
Image Credits Nofollow v.1.3 by Apasionados.es
Jetpack von WordPress.com v.7.6 by Automattic
WPBakery Page Builder v.6.0.5 by Michael M - WPBakery.com
LEVELUP Core v.1.1.37 by Puzzlerbox
Loco Translate v.2.3.0 by Tim Whitlock
Mailchimp for WordPress v.4.5.3 by ibericode
New RoyalSlider v.3.3.5 by Dmitry Semenov
WP No External Links v.1.0.2 by nicolly
No Follow All External Links v.2.3.0 by gearpressstudio
Virtual Robots.txt v.1.9 by Marios Alexandrou
PHP Code Widget v.2.3 by Otto
Publicize With Hashtags v.0.1.4 by Darkseal
Really Simple CAPTCHA v.2.0.2 by Takayuki Miyoshi
Regenerate Thumbnails v.3.1.1 by Alex Mills (Viper007Bond)
Relevanssi v.4.3.2 by Mikko Saari
Remove Google Fonts References v.2.6 by Bruno Xu
Remove Yoast SEO Comments v.3.1 by Mitch
Slider Revolution v.5.4.8.3 by ThemePunch
Search In Place v.1.0.42 by CodePeople
SEO Friendly Images v.3.0.5 by Vladimir Prelovac
WP DSGVO Tools (GDPR) v.2.2.19 by Shapepress eU
Sitemap v.4.3 by webvitaly
Theme My Login v.7.0.14 by Theme My Login
Envato Toolkit v.1.4 by KestutisIT
User Role Editor v.4.51.2 by Vladimir Garagulya
Video Thumbnails v.2.12.3 by Sutherland Boswell
Yoast SEO v.11.9 by Team Yoast
WP Rocket v.3.3.6 by WP Media
WP Live Search v.0.9 by 
WP MU plugins	
WP Migrate DB Pro Compatibility v.1.1 by Delicious Brains

Meanwhile I read this thread and it seems to be the same problem. Images are not loaded and the directory has not been created. Could it be a problem of my webhost? Thanks a lot!