Possibly you need to put my state of confusion down to advancing years, or yesterday's session with the dentist (from which I am still recovering).
There are lots of terms being used, which you haven't given a context for, so I'm not sure where your confusion comes from.
Does your connection go:
Internet->Thomson TG585->Netgear VMDG280->rest of local appliances?
If so, then the internet IP 81 ... will get to the Thomson (leaving aside the method by which it gets there) quite naturally.
I use 10.0.0.138 as my modem/router local facing address - purely because my original equipment used this by default, and I continue it to avoid my own confusion.
If you change the 'gateway' address (however you define this in your own mind) you need to reboot PCs using DHCP to allow them to use the new address (unless your more experienced than your posts give me to understand).