Fixed! The problem was that on the online server, the application folder is placed OUTSIDE the public_html folder. Therefore the public_path() was pointing at the "wrong" folder.

# root
## laravel
## public_html
### images
#### users

Therefore the application was uploading the images into

# root
## laravel
### public
#### images

I managed to resolve this with a quick fix in the controller.

if ($request->file('profile_pic')->isValid()) {
    $path = base_path();
    $path = str_replace("laravel", "public_html", $path); // <= This one !
    $destinationPath = $path . '/images/users'; // upload path
    $extension = $request->file('profile_pic')->getClientOriginalExtension(); // getting image extension
    $fileName = uniqid() . '.' . $extension; // renameing image
    $request->file('profile_pic')->move($destinationPath, $fileName); // uploading file to given path
    $user->profilepic = $fileName;
}