Your old DS must have had FlashMe installed, while your new DS Lite doesn't. You need to trick the DS into booting into DS mode from the GBA cart, which can be done with a slot-1 NoPass device (like the Superkey or PassCard 3). Once you've done that at least once, you can install FlashMe, which is a replacement firmware that is capable of booting into DS mode from the GBA slot without a NoPass device.