Wordpress - Get first video from the post (both embed and video shortcodes)

You can use the get_media_embedded_in_content() function for both shortcode and embedded media (it looks for these HTML tags: audio, video, object, embed, or iframe), just be sure to use content with applied filters and shortcode executed.

For example:

$post_id = 125;
$post = get_post($post_id);

//Get the content, apply filters and execute shortcodes
$content = apply_filters( 'the_content', $post->post_content );
$embeds = get_media_embedded_in_content( $content );

//$embeds is an array and each item is the HTML of an embedded media.
//The first item of the array is the first embedded media in the content
$fist_embedded = $embeds[0];

Addapting te above code to a function similar to yours:

function get_first_embed_media($post_id) {

    $post = get_post($post_id);
    $content = do_shortcode( apply_filters( 'the_content', $post->post_content ) );
    $embeds = get_media_embedded_in_content( $content );

    if( !empty($embeds) ) {
        //return first embed
        return $embeds[0];

    } else {
        //No embeds found
        return false;
    }

}

To limit videos only you could do something like this:

function get_first_embed_media($post_id) {

    $post = get_post($post_id);
    $content = do_shortcode( apply_filters( 'the_content', $post->post_content ) );
    $embeds = get_media_embedded_in_content( $content );

    if( !empty($embeds) ) {
        //check what is the first embed containg video tag, youtube or vimeo
        foreach( $embeds as $embed ) {
            if( strpos( $embed, 'video' ) || strpos( $embed, 'youtube' ) || strpos( $embed, 'vimeo' ) ) {
                return $embed;
            }
        }

    } else {
        //No video embedded found
        return false;
    }

}

Note: There was a bug in get_media_embedded_in_content() that made this answer not working as expected bellow WP 4.2.

You could try mashing all the patterns together in one big regexp or and then doing a simple line by line parse of the content:

function theme_oembed_videos() {

    global $post;

    if ( $post && $post->post_content ) {

        global $shortcode_tags;
        // Make a copy of global shortcode tags - we'll temporarily overwrite it.
        $theme_shortcode_tags = $shortcode_tags;

        // The shortcodes we're interested in.
        $shortcode_tags = array(
            'video' => $theme_shortcode_tags['video'],
            'embed' => $theme_shortcode_tags['embed']
        );
        // Get the absurd shortcode regexp.
        $video_regex = '#' . get_shortcode_regex() . '#i';

        // Restore global shortcode tags.
        $shortcode_tags = $theme_shortcode_tags;

        $pattern_array = array( $video_regex );

        // Get the patterns from the embed object.
        if ( ! function_exists( '_wp_oembed_get_object' ) ) {
            include ABSPATH . WPINC . '/class-oembed.php';
        }
        $oembed = _wp_oembed_get_object();
        $pattern_array = array_merge( $pattern_array, array_keys( $oembed->providers ) );

        // Or all the patterns together.
        $pattern = '#(' . array_reduce( $pattern_array, function ( $carry, $item ) {
            if ( strpos( $item, '#' ) === 0 ) {
                // Assuming '#...#i' regexps.
                $item = substr( $item, 1, -2 );
            } else {
                // Assuming glob patterns.
                $item = str_replace( '*', '(.+)', $item );
            }
            return $carry ? $carry . ')|('  . $item : $item;
        } ) . ')#is';

        // Simplistic parse of content line by line.
        $lines = explode( "\n", $post->post_content );
        foreach ( $lines as $line ) {
            $line = trim( $line );
            if ( preg_match( $pattern, $line, $matches ) ) {
                if ( strpos( $matches[0], '[' ) === 0 ) {
                    $ret = do_shortcode( $matches[0] );
                } else {
                    $ret = wp_oembed_get( $matches[0] );
                }
                return $ret;
            }
        }
    }
}

Also you might want to use transients for the embeds - see this answer I gave.