Rewriting Hostname & URI on HTTP_Request without a 'redirect'

You can set the uri and host directly on a request, and forward to a configured pool internal to the system without issuing a redirect:

when HTTP_REQUEST {
  if {[HTTP::uri] starts_with "/index.php"} {
    HTTP::uri "/app1[HTTP::uri]"
    HTTP::host "www.website2.com"
    pool your_pool_name
  } else {
      HTTP::redirect "https://[HTTP::host][HTTP::uri]"
  }
}

You might want to include additional checks in your if statement to make sure the site is www.website1.com, depending on your requirements.