The "augmented" label confused me a lot too. I saw Steve Thomas's presentation in person at DEF CON and could not find material online using that phrase outside of his materials.
Sc00bz made a comment[1] on reddit describing what "doubly augmented" means where they succinctly describe it as
"A doubly augmented PAKE is where both sides can store data to authenticate with eachother but not themselves."
This mirrors the definition of aPAKE and makes sense, but this sounds more like the security property provided by CRISP[2]. This doesn't make sense in the context of double BS-SPEKE as the client inputs their password in the clear which can be used to generate the server's storage without any "guessing".
[1] https://old.reddit.com/r/crypto/comments/zy8w3y/what_we_do_i...
[2] CHIP and CRISP: Protecting All Parties Against Compromise through Identity-Binding PAKEs https://eprint.iacr.org/2020/529
" Client without password has:
idC = client identity
P = hashToPoint(p)
c
S = s * P
Reg = reg * G (reg = registration private key)
"