I did try using dictionaries, but unfortunately, they were, at least to me, of little help, as most of them defined the two words similarly.
Their meanings are somewhat similar but they are used rather differently.
"Disheartened" is usually used to describe a person. That is, a person who has lost hope, lacks drive and so on is disheartened.
"Half hearted" is usually used to something that a person does.
"Joe was disheartened because of the many disappointments he had had in his life".
"Jill's attempts to play piano were half-hearted and that is one reason she never played well."