Hopefully seeing instructions used in-place makes it a bit clearer. Note the instruction sequence for AES128/192/256 is slightly different (i.e. each of the SAES_KEY*_STEP macros is different).
Barry's original proposal for the key schedule instructions on RV64 can be found here:
https://lists.riscv.org/g/tech-crypto-ext/message/126.
Page 17 of the attached PDF onwards describes the "new key schedule proposal". This is a
very old discussion, but the rationale for the keyschedule instructions I think remains exactly as described here.
I hope this helps? If not let me know and I'll try to fill in any more gaps.
Cheers,
Ben