How do you get an interest-free conditional debt recognized? Let's say I sign a contract with a relative that says upon my death, you get X amount of money if I have enough money to service that debt, but there's no interest rate and if I have less money than that you just get all of my money. How do I get a debt written in a contract recognized by financial institutions? Are those even considered debts legally speaking? You can assume that the country is the U.S.
Let's say you have an actual contract where you give A something of value in consideration of which B gives you something of value. That way, the exchange is legally enforceable and is not just a cancellable promise. This is all written up clearly, and you become a creditor. One thing to bear in mind is that you may have to wait in line so that others are paid first. For example, taxes have to be paid, also secured creditors have to be paid. Nearly at the end of the list, unsecured creditors such as you are paid, and finally heirs can receive shares of what is left.
I don't know what you mean by "recognizing" a debt, but what a financial institution could do is treat a debt as an asset, for the purposes of a loan. For instance, a bank might lend you $10,000 where you pledge your home as a thing of value that they can take in case you default on repayment. You cannot force a bank to accept a particular asset as collateral, so the answer to the "how" question is "by persuasion". Since your asset has an indefinite value between $0 and let's say $10,000, you would have to persuade the bank that there is negligible risk to them. In the case of property used as collateral, when you default the creditor can use the courts to seize your property and sell it, but they can't seize the property of a third party who owes you something, they would have to wait possibly for years for the third party to die and see what's left. Investment instruments might be used as collateral, but only if they can be liquidated reasonably promptly. The risk to the bank is extremely high, but if you are very persuasive, they might accept such a debt as collateral.