Scheduling a broadcast to record remotely

Do I have it correct that I can only do this on a previously connected device such as a tablet or etc.? Am I not able to log in to some sort of web interface from outside my network to schedule a recording?

That is correct.  You can connect to your Tablo from outside your home network, but only from a device that has been paired with the Tablo on the home network previously.

There is a chance that the Tablo folks will change that in the future since it has been requested several times.