A customer generally provides requirements (the system should do...) which are translated into a spec (the module/function/method should do...). The set of specs map to requirements. Requirements may be derived from or represented by user stories and specs may or may not by developed in an agile way or written down ahead of time. Whether you have or derive requirements and specs is entirely orthogonal to development methodology. People need to get away from the idea that having specs is any more than a formal description of what the code should do.
The approach we take is the specs are developed from the tests and tests exercise the spec point in its entirety. That is, a test and a spec are semantically synonymous within the code base. Any interesting thing we're playing with is using the specs alongside the signatures to have an LLM determine when the spec is incomplete.
A spec consists of three different kinds of requirements: functional requirements, non-functional requirements, and constraints. It’s supposed to fully describe how the product responds to the context and the desires of stakeholders.
The problem I see a lot with Agile is that people over-focus on functional requirements in the form of user stories. Which in your case would be statements like “X should do…”
I always find the distinction between the two fuzzy (because many non-functional requirements can be argued to be functional requirements) but the list here is useful for the discussion: https://en.wikipedia.org/wiki/Non-functional_requirement
Take things like "capacity". When building a system, you may have a functional requirement like "User can retrieve imagery data if authorized" (that is the function of the system). A non-functional requirement might be how many concurrent users the system can handle at a time. This will influence your design because different system architectures/designs will support different levels of usage, even though the usage (the task of getting imagery to analyze or whatever) is the same whether it handles one user at a time or one million.
The approach we take is the specs are developed from the tests and tests exercise the spec point in its entirety. That is, a test and a spec are semantically synonymous within the code base. Any interesting thing we're playing with is using the specs alongside the signatures to have an LLM determine when the spec is incomplete.