The difference comes between the substitution of a natural right for a legal right. A natural right is inalienable, all people are born with it regardless of their citizenship or status. A legal right is something that a government bestows on certain people. You can't have a natural right that compels action in someone else.
A natural right is something that everyone has that they are born with and something that can only be taken away. Natural rights are things like the freedom to speak without being censored. The freedom to think or believe without being oppressed. The freedom to own and defend personal life and property. The freedom to have a fair legal hearing. Most of the bill of rights derives from these rights. Example, the second, third, and fourth amendment are specific ways to exercise the right to personal defense and property rights. The fifth, sixth, seventh, and eighth are specific applications of a right to a fair trial.
A natural right might be the freedom to travel freely. A legal right would be the right the state bestows upon you to drive.
You really don't have the natural right to an attorney in the US, you have the right to have legal representation. Much like the natural right to free speech doesn't mean that you get to have TV time whenever you want. That language has been misinterpreted which creates the current system of free public representation (a legal right).
Same for healthcare. In order for healthcare to be a natural right, you obligate someone to work in healthcare. Instead, it is a legal right that you can't be turned away from a hospital if you need medical attention.
Now people will interchange them and put them on equal footing, but they aren't. As you correctly assumed, you can't have a natural right to something that doesn't necessarily exist.