That local IP (192.168.x.x) which you get from your router is not what will be seen from the outside
(what that website will see).
The outside world will see the IP of your router.
You usually get that assigned dynamically by your ISP - rebooting the router or disconnecting/reconnecting the power may get you a different address - or not.
It depends on your ISP and how they assign addresses to their customers.
Changing or forcing the router to assign you a different local IP (192.168.x.x) will do nothing with respect to the outside (the internet).