Why is the text displaying, but not the textarea field itself?

This is a PHP syntax issue, you can’t put a ternary operator inside an echo. Assign the value to $meta in one line:

$meta="" == get_post_meta($post->ID, 'twitter_embed', true) ? get_option('my_option_name') : get_post_meta($post->ID, 'twitter_embed', true);

Hata!: SQLSTATE[HY000] [1045] Access denied for user 'divattrend_liink'@'localhost' (using password: YES)