Engaged, a.
1. Occupied; employed; busy.
2. Pledged; promised; especially, having the affections pledged; promised in marriage; affianced; betrothed.
3. Greatly interested; of awakened zeal; earnest.
4. Involved; esp., involved in a hostile encounter; as, the engaged ships continued the fight. Engaged column. (Arch.) Same as Attached column. See under Attach, v. t.
Shades of meaning
Fernald has no entry of its own for engaged; he lists it under industrious.
Where he writes about it:
One is bound by conscience or honor; he is obliged by some imperious necessity; engaged by his own promise; compelled by physical force or its moral equivalent. — from bind
It is unsuitable, according to present usage, to speak of hiring a pastor; the Scripture, indeed, says of the preacher, "The laborer is worthy of his hire;" but this sense is archaic, and hire now implies that the one hired works directly and primarily for the pay, as expressed in the noun "hireling;" a Pastor is properly said to be called, or when the business side of the transaction is referred to, engaged, or possibly employed, at a certain salary. — from employ