The three terms really do convey essentially the same meaning. "Nurture" is somewhat more intimate, perhaps best suited to someone close to you (or yourself):
My parents, both public defenders, nurtured my respect for fairness.
To talk about "cultivating" people is mildly insulting, as people are not crops, but it would be appropriate to say something like
I intend to cultivate an environment of honesty in this classroom.
"Foster," or to promote, is neutral and would also be appropriate for a business
or teaching environment:
A manager's first duty is to foster integrity.
Another associated word that you might find useful is "instill," or to introduce or implant:
I exposed you to those challenges to instill a sense of bravery in you.
As you note, "nourish" may not be as well suited because qualities such as honesty are not in themselves nourishing (although your attention and encouragement could very well be emotionally nourishing).