Poor but not financially, rich but not through money
Solution 1:
It sounds as though you are trying to write about the difference between wealth and nobility.
One who is poor financially is in poverty, or economically destitute, or bankrupt, though the last one is also a technical term. One who is poor in ethical qualities could be called inethical, immoral, abhorrent, crude, or any other number of words to describe one who is poorly-behaved. VERY specifically, if it is someone who doesn't share their money, you could say they are "cheap" or "a miser" or, more kindly, a "spendthrift" if they always spend as little as possible on their ventures.
Metaphorically, one who has no moral guidance or standards can also be called 'Bankrupt', though usually it is specifically called 'morally bankrupt' - meaning this would be two words, not one.
Financial richness has quite a few terms to describe it: wealthy, well-off, well-to-do, or one could be a millionaire, billionaire, or even trillionaire (All three being categorical). Whereas being rich in an ethical sense would be honorable, noble, decent, kind-hearted, and willing to share wealth would be generous, philanthropic/philanthropist, and charitable.
I believe the term you're looking to use depends on the exact nature you are trying to express in the person you're describing. Are they very willing to offer their wealth, even if they have little? Are they individuals who are incredibly wealthy, but who refuse to hand out their wealth? Or is it the grace of the person that you are judging? As in a poor individual who acts with dignity, or a wealthy individual who acts very cruel?
There is certainly a multitude of metaphors connecting wealth with virtue. Even "noble" has the connotation of being one of a higher-class and, therefore, wealthier than others (though this is a connotation, and in actuality nobles are not always wealthy). Rich in virtue, poor in manners, there's quite a lot of ways you can phrase what you want to say.
Solution 2:
Jesus said it fairly well in a couple places in His Sermon on the Mount and elsewhere:
"Blessed are the poor in spirit, for theirs is the kingdom of heaven" (Matthew 5:3).
In other words, people are rich when they recognize how truly impoverished they are in light of what they could be and are not, but would like to be. Put another way, people who have a humble estimation of their accomplishments are richer, ironically enough, than the folks who think they're just fine as they are, thank you very much!
"'Beware, and be on your guard against every form of greed; for not even when one has an abundance does his life consist of his possessions'" (Luke 12:15).
And just a few verses in Luke Chapter 12, Jesus has this to say,
So is the man [a fool] who stores up treasure for himself, and is not rich toward God (Luke 12:21 NASB).
In other words, a truly rich life does not consist merely of having lots of stuff. That's where we need to distinguish between needs and greeds. Contentment with having the necessities of life (i.e., our needs are met) is far better than discontentment with never having enough (i.e., our greeds are never met)!
In conclusion, I would "translate" your four terms as Jesus would from his perspective:
- Poor financially: Jesus's term for this would be "poor," literally, meaning possessing very little stuff
- Poor non-financially: Jesus's term for this would be "deluded" or a "fool," since financial wealth, or an abundance of stuff, can become an idol, which is a God substitute and is worthless when death comes a knockin' at the door
- Rich financially: Jesus's term for this would be "rich," literally, meaning having lots of stuff
- Rich non-financially: Jesus's term for this would be both "poor in spirit" and "rich toward God."