How to change a Vimeo's video privacy via API (PHP)?

I'm trying to set a Vimeo's video privacy via API (v3.4) request.

Sending an HTTP PATCH to https://api.vimeo.com/videos/{videoId} with

Headers:

"Authorization" => $api_key,
"Accept" => "application/vnd.vimeo.*+json;version=3.4"

Body:

"form_params" => array(
    "privacy" => array(
        "embed" => "public"
     ),
     "name" => $video_name,
     "description" => $video_description
 )

...properly changes the video's name and description (the API token has edit permission), but the privacy setting remains untouched.

I've followed every step specified by the Vimeo's API Documentation but I can't get it to work. What am I doing wrong?

The privacy field is actually privacy.{key}. So, the correct code is

"form_params" => array(
    "privacy.embed": "public"
    "name" => $video_name,
    "description" => $video_description
)