How to Manage Comment Notification Emails in WordPress
Let’s take a look at the “Email me whenever” option. These two control when you receive a message regarding the comment section. By default, WordPress will have both options activated. This means you’ll receive an email when someone posts a comment or if a comment is held for moderation.
ADDED: passing through the $notify variable, available sinds 4.6. This is for other plugins to override default sending to admin or user. Only useful if sending within this plugin is activated.
UPDATED: updated with the newer pluggable send functions of wordpress 4.7.
FIXED: Missing blogname in user email