I wasn't aware I was running a modem/router!
How do I confirm either, and then how do I disable for a reboot?
LOL - you must be, to get online! Very old Macs included a modem which would get online using dial-up/phone line. These days the modem (for broadband) is contained within a router which is typically : a flat case about the size of a small paperback with an array of lights, usually green when all is working, one or two of which will be flashing on and off when connected.
This router will usually have a power on/off button - if so, close your browser & email and turn it off.
If not, and you're connected by Ethernet (yellow cable into your Mac) - " " " " then detach cable from Mac
If not, and you're connected by WiFi, " " " " " , then go to the fan-shaped icon in your menu bar and turn Airport off.
You will then be offline.