Natural rights and human rights originally come from different vocabularies. It's not fair to construe natural rights as "simply a less developed and more concise version of what we now consider human rights."
First, I want to start by pointing out an important but crucial ambiguity in the term "natural rights." Viz., the problem is that "nature" can mean many things. Thus, one meaning of natural rights are those rights granted to an individual by Nature (though in naturalized versions not capitalized); another meaning is those rights granted to a person as a result of their nature. Thus, the US Declaration of Independence speaks of natural rights as those granted by the Creator in making humans the way they are. The second meaning of natural right could be stated in saying people have rights to food, community, and procreation, because these follow from human nature. The senses are somewhat integrated in that most rights granted by Nature (as God, Nature, nature) are generally presumed to accrue as consequences of the sort of creatures that are made.
To make it simpler with an analogy, if God declares we have the right to free tacos, then this would be natural in the first but not the second sense, but if God makes us so that we need food, then we may have a natural right in the second sense to the provision of food [thought not necessarily tacos].
Human rights can differ from both of these senses or overlap. Here, the adjective identifies those things which accrue to humans (an objective genitive) rather than those things that follow from being human. Thus, any right that we think people deserve and/or we grant to people is a human right if we grant it to them because they are human.
To give an example, the UN declares that all humans have the right to a country of citizenship. This can be understood in one of two ways. Either you could believe this follows from the nature of human beings and is thus a human right we should recognize or you believe that this is a right we have given humans by legal force and thus becomes a human right.
Again, we can return to our example. If the UN council passes a bill saying we deserve free tacos, then we have a human right to free tacos in those places where UN pronouncements mean something. The UN might argue that these rights follow logically from what we are in which case they are claiming a natural right-basis for this human right or rather that they are merely pronouncing an already existing natural right and codifying it.
We can now return to the sense in which you viewed human rights as progress over natural rights. This is definitely there in the literature. But this does not mean human rights are necessarily better. What it means is that our idea of rights has shifted. When rights talk first takes off in the natural rights form, it is seen as the perception of a pre-existing right inherent in nature (in the second sense) or a right God/Nature has woven into the fabric of existence (first and second senses together). Human rights talk can also include positive rights -- rights that exist only because we've decreed that they do. But with this comes the critiques.
Human rights can wind up being arbitrary precisely insofar as they are not necessarily grounded in nature. They also tend to be revocable or if rights with no basis are granted, deleterious to a society. Moreover, this type of rights talk requires a certain Western type of legal system. Thus, we see many critiques from Chinese philosophy (see Confucian Ethics: A Comparative Study of Self, Autonomy, and Community and the work of Theodore de Bary) and from Western communitarians (e.g. Alasdair MacIntyre) and from feminists sand others.