I think you got the point splendidly, Theseus. :)
Διάβασμα also means studying. So, when a parent says «διάβασε» or asks «τι έχεις να διαβάσεις για το σχολείο;», she means "study" not simply "read".
The «δεκάξι, μεκάξι, ξεκάξι» part (a similar, more free in its form -- you could select any...