Oh and at the end you said ".....the dependent is an unrelated disabled person" . That is why you aren't head of household. They have to be related to you.
Who is a qualifying person for Head of Household?
https://ttlc.intuit.com/community/family/help/what-is-a-qualifying-person-for-head-of-household/00/2...
But for getting the dependent credit you misread that question if THEY paid more than half. See the screen shot posted above.
