Send a Receipt to a user when adding a manual membership OR any payment
You may be able to hook something together using CiviRules, but it's unlikely - because CiviRules won't have the same context for generating the receipt.  So CiviRules is probably fine for sending an internal email stating, "Hey, this contribution came in, here are the details", but (most likely) can't generate the exact receipt.

That's probably good enough - but if it's not, you can create an extension to handle this, using hook_civicrm_alterMailParams.