He won't be able to attend because he's 'otherwise engaged'. [=he is busy doing something else]
('US') He is 'engaged' in research. = ('Brit') He is 'engaged' on research.
She 'engaged' him 'in' conversation. [=she started a conversation with him]
(..) If you had any cultural inclinations this evening. You have a grand gift of silence, Watson, it makes you quite invaluable as a companion. I knew she'd been engaged. She told me. So that's, no to the opera then. That was my waistcoat. I thought we agreed it's too small for you. (..)2009 Sherlock Holmes
(..) Pearls, diamonds, flawless rubies. Hardly the gems of a governess . However, the jewls you are not wearing, tell us rather more. - Holmes! - You were engaged. The ring has gone, but the lighter skin suggests that you spent some time abroard proudly that is until you were (..)2009 Sherlock Holmes