When sending an email to a ticket using [#INC-123] in subject line, allow the note created to be marked as private or public from the subject line using a token like [X/0] or even use an exclamation mark ! to set as Private or nothing for default action of public note
Example
[#INC-123][x] - to send a private note to ticket 123.
OR
[#INC-123!] - to send a private note to ticket 123


