There are differences in the OFW for the hardware difference but the functionality like flashcart blocking is still functionally the same. There’s a lot of technical details but they really aren’t that important to the average user
I am pretty sure they do not but in any case Luma CFW bypasses the flash cart blacklisting which means any cart will work except ones that rely on the original DS boot sequence like the original R4 but even those can be booted with a Stage2 launcher. In otherwords, if it worked on DSi or 3DS firmware 1.0.0 it will work with Luma.
That being said, the DSTwo is known to have issues with fitting and making contact properly because it is thicker than other carts so it is most likely a physical issue, not a firmware issue.