Congratulations! You're on the right track to finding the secret button page but you're not there yet. You've made it this far but now there's more you have to do before you can get there. You have to...
For the true Fire Cabin experience, play as the Farmer from Illinois with Vuur, Maurice, Aether, Cordelia, and Kor in your party.
Just in case the embed acts up: Oregon Trail Game Online | Play Free Fun Retro Internet Games (internetgames365.com)
And remember where this page is because you'll have to come back here later ;)