Use an appropriate name in the Reply-To header

Use "Site Name <>" for the Reply-To header when the reply is to the site or a public topic.

Use "username <>" for the Reply-To header only when the reply is to a private message topic.

app/mailers/user_notifications.rb

@@ -215,6 +215,7 @@ class UserNotifications < ActionMailer::Base
       username: from_alias,
       add_unsubscribe_link: true,
       allow_reply_by_email: allow_reply_by_email,
+      private_reply: post.topic.private_message?,
       include_respond_instructions: !user.suspended?,
       template: template,
       html_override: html,

lib/email/message_builder.rb

@@ -136,6 +136,13 @@ module Email
       @opts[:allow_reply_by_email]
     end
 
+    def private_reply?
+      SiteSetting.reply_by_email_enabled? &&
+      reply_by_email_address.present? &&
+      @opts[:allow_reply_by_email] &&
+      @opts[:private_reply]
+    end
+
     def from_value
       return @from_value if @from_value
       @from_value = @opts[:from] || SiteSetting.notification_email
@@ -149,7 +156,11 @@ module Email
 
       @reply_by_email_address = SiteSetting.reply_by_email_address.dup
       @reply_by_email_address.gsub!("%{reply_key}", reply_key)
-      @reply_by_email_address = alias_email(@reply_by_email_address)
+      @reply_by_email_address = if private_reply?
+                                  alias_email(@reply_by_email_address)
+                                else
+                                  site_alias_email(@reply_by_email_address)
+                                end
 
       @reply_by_email_address
     end
@@ -159,6 +170,10 @@ module Email
       "#{@opts[:from_alias]} <#{source}>"
     end
 
+    def site_alias_email(source)
+      return "#{SiteSetting.title} <#{source}>"
+    end
+
   end
 
 end