Filter a few parameters into WordPress YouTube oEmbed requests. Enable modest branding which hides the YouTube logo. Remove the video title and uploader information. Prevent related videos from being shown once the video has played.

functions.php

<?php
/**
 * Filter a few parameters into YouTube oEmbed requests
 *
 * @link http://goo.gl/yl5D3
 */
function iweb_modest_youtube_player( $html, $url, $args ) {
	return str_replace( '?feature=oembed', '?feature=oembed&modestbranding=1&showinfo=0&rel=0', $html );
}
add_filter( 'oembed_result', 'iweb_modest_youtube_player', 10, 3 );

In case someone else comes across this and wants to change the settings for youtube embeds in Wordpress, this only works when you're adding a new video to a page not on existing embeds. In addition, this doesn't work with the way YouTube gives share URLs which uses the youtu.be domain name. This is how I altered it:

function imp_custom_youtube_querystring( $html, $url, $attr, $post_id ) {
  if (strpos($html, 'youtube')!= FALSE || strpos($html, 'youtu.be') != FALSE) {
    $args = [
      'rel' => 0,
      'controls' => 0,
      'showinfo' => 0,
      'modestbranding' => 1,
    ];
    $params = '?feature=oembed&';
    foreach ($args as $arg => $value) {
      $params .= $arg;
      $params .= '=';
      $params .= $value;
      $params .= '&';
    }
    $html = str_replace( '?feature=oembed', $params, $html );
  }
  return $html;
}
add_filter('embed_oembed_html', 'imp_custom_youtube_querystring', 10, 4);

One thing you should note about this is that it will run on every embed on every page load rather than have the more performant way that is suggested above. Probably not that big of a deal unless you've got a really high volume site though, in which case you're better off just removing and re-embedding the video and using the oembed_result filter from above instead.