Advancing the corpus linguistics, looking specifically at tragedy, the emphasis is on banish and banishment because of the political context.  Exclude/exclusion are not encountered at all.  (Corpora obtained from ProjectGutenberg and run through TextSTAT).

Duchess of Malfi: banished x5; banishment x2. Inc. ANTONIO: ‘My banishment, feeding my melancholy…’

The White Devil: banish/ed 3; banishment 3.  Opening with Ludovico’s exclamation ‘Banished!’.

Although the term is not encountered in A Woman Killed with Kindness, banishment is effectively the fate of Anne (Nan).

Banishment was the sense invoked by urban authorities in their exclusion of their own members who had allegedly contravened expected norms.