Yes, unhorsed knights are an exception, because they are not crusade units per se (they don't have the crusading tag).

You cannot hire pilgrims, fanatics, crusader knights, crusader sergeants, flagellants and great crosses outside of crusades though, and also cannot retrain those.

Units added to a stack become crusaders by default under following conditions: They are already crusaders, they were in the stack when general joined crusade, they are in crusade stack at end of turn and outside a settlement.
If you add a unit to an exisiting stack without rejoining the crusade with a general, they won't be crusaders till the next turn.
That's hardcoded.