I think the working space in a given chat or session still counts as mind enough, but your argument is different, in that it argues about specific architectures rather than simply stating "software therefore cannot be conscious".
Fwiw, I interpret poster above as referring to root of the word. The English root for mind has a strong connotation of requiring memory (and commonly in other IE languages).
In western philosophy, say in aquinas, the mind is something that can discern moral truths and reason about morality. It is "soulful".