What does Dimitri mean?
Dimitri honours Demeter, the Greek goddess of the harvest and earth.
What does Basil mean?
From Greek 'basileios', meaning 'kingly, royal', derived from 'basileus' ('king'). Carried by Saint Basil the Great. The sense is the adjective 'royal/kingly' rather than the noun 'king'. (The herb shares the same root.)