Updating the game required an updated firmware. While you did update the game on another 3DS, said update will only work on that 3DS. On your non-updated 3DS, where you didn't have a game update, it will stay as that. Therefore you can't go online in that 3DS, but can use the browser exploit. Whereas in the updated 3DS, you can go online, but you can't use the browser exploit.
I understood the situation after reading it a few times. The console and the native SD card carry the update, NOT the cartridge. If you updated the game through the non-updated 3DS, it won't work. If you just switched consoles, then you're fine.