different between detachment vs disengaged

English

From French détachement; synchronically analyzable as detach +? -ment.

detachment (countable and uncountable, plural detachments)

(uncountable) The action of detaching; separation.
(uncountable) The state of being detached or disconnected; insulation.
(uncountable) Indifference to the concerns of others; disregard; nonchalance; aloofness.
(uncountable) Absence of bias; impartiality; objectivity.
(uncountable, military) The separation of a military unit from the main body for a particular purpose or special mission.
(countable, military) The unit so dispatched.
(countable, military) A permanent unit organized for special duties.
(countable) Any smaller portion of a main body separately employed.

From disengage +? -ed.

disengaged

disengaged (comparative more disengaged, superlative most disengaged)

Unconnected; detached.
(dated) Not (socially) engaged; available, free.
- 1890, Oscar Wilde, The Picture of Dorian Gray, Vintage 2007, p. 40:
  ‘You must come and dine with us some night. Tuesday? Are you disengaged Tuesday?’

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