There are three separate things conflated here:
- The protocol (the SD Physical Layer specification that applies - now up to v6.0) implemented by the SD to NAND flash controller chip
- The amount of NAND flash memory attached to the controller chip.
- The factory formatting of the card (FAT16/FAT32/exFAT)
The SD/SDHC/SDXC label is marketing that simplifies the above. The main differences are:
- SD: the original. FAT16 formatted. Can be up to 4GiB but in practice usually 2GB is the upper limit.
- SDHC: 4-32GiB. Some protocol changes (sec 4.2.3 and 4.3.14). Typically needs firmware changes (eg in a USB card reader) to support. FAT32 formatted. Most 4GB cards are SDHC, even though the spec says SD can be up to 4GiB.
- SDXC: exFAT formatted. No protocol changes I could see in the spec. Cards above 32GiB are SDXC.
In other words the hardware/firmware is the same for SDHC and SDXC, it's just the pre-written data that's different.
Now, there's no reason you couldn't fit a 2GiB flash chip to a SDHC controller, format it as exFAT and call it SDXC. You lose the ubiquity of FAT (users of exFAT have to pay a patent fee to Microsoft, hence it isn't so common) but the extra features of exFAT may make up for it. The SD Association might complain about confusing marketing if you tried to sell it, though.
That said, the formatting of the card is just a case of the data written to it in the factory. If you reformat a 2GB SD card to exFAT you achieve almost the same effect. Arguably it's better since it will work in some old card readers and such whose firmware doesn't support the SDHC protocol.