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.
2 Answers
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.
So:
"Joe was disheartened because of the many disappointments he had had in his life".
vs.
"Jill's attempts to play piano were half-hearted and that is one reason she never played well."
There is a difference in meaning: disheartened means you feel disappointed and discouraged, whereas half-hearted means showing no real effort, interest or enthusiasm. You only put half of your heart in so to speak.