AFAIK, email signatures are specific to the program you use to send the email.
E.g. if I set up a signature for my Gmail account which appears every time I send mail using the Gmail web interface, the same signature won't appear if I use Thunderbird to send messages from the same Gmail account.
I would have to set it up once for Gmail and once for Thunderbird.
What you can certainly do is set three signatures, one for each of the cases you list. This will work.
However, I'm not a HostGator customer, so it might be worth contacting their support, just to find out for sure if a unified signature is possible.