The first time I gave considerable thought to this question was when I was working at the Israel Antiquities Authority, attempting to collaborate with Google to create the first version of the “Leon Levy Digital Dead Sea Scrolls Library” website, which launched in December 2012:
https://www.deadseascrolls.org.il/home

I had no formal theoretical background in structuralism or post-structuralism, and had only passing familiarity with such concepts as the death of the author, or approaches such as “reader-response”. But I had a practical problem on my hands. I had to try to explain to Google that it would be unhelpful to simply apply the label “Genesis” as an identifier for all the hundreds of fragments that remained from a couple of dozen different copies of the book of Genesis, that had been recovered from 3 different caves at Qumran and from the site of Masada. Each of those fragments would need a label, and each of the original scrolls that they came from needed a label too–something that would show that each fragment, and the presumed “original scroll” that a set of fragments came from were physical artefacts.

It was also still important to me to designate the literary content inscribed on those physical artefacts, for example, the Book of Genesis. And it mattered that the text of the Book of Genesis was sometimes slightly different in a fragment from one scroll than it was in a fragment of another, and that sometimes these “readings” were “witnesses” to a different text than the traditional Hebrew or Greek texts of the Bible. I spent a lot of time worrying about what the words I wanted to use “really” meant, but my primary interest was to communicate effectively with my colleagues at the Antiquities Authority, with the Google team, and especially with the future users of the website. I needed to assign practical definitions, and to use terms consistently.

In the end, this is what my colleagues and I came up with, and placed in the “help” text of the website (which I suspect never got much traffic. But the enterprise of thinking about this together with some other colleagues in the field of Dead Sea Scrolls Studies did seem to have some impact on notions of “textuality” in the field, and dovetailed with growing interest in material philology):
https://www.deadseascrolls.org.il/help

What is the difference between a text, a manuscript, and a composition?

A text is a unit of writing. It could be a hymn, a legal contract, or an inscription on a clay potsherd. A manuscript is a physical object, the material form of a recorded text. Only about a dozen of the Qumran Cave Scrolls were recovered intact as complete scrolls. Most manuscripts survived only in fragments. Scholars sorted all of these fragments on the basis of physical material and shape, content, and handwriting in order to reconstruct the original manuscripts when possible. Each Dead Sea Scroll manuscript is unique, even if its content is also found in other manuscripts. A composition is a specific literary or documentary entity, such as Genesis or a Deed of Sale, which has been recorded on a manuscript. A composition may be preserved in a single copy among the Dead Sea Scrolls, or found in several manuscripts.