As a body of a suitable size and mass moves within the Roche limit of a planet, it will disintegrate. This will form a debris field that will eventually form a planetary ring. Is there a name given to this debris field before it forms a ring? Something like a Roche train or Roche field seem appropriate but I can't find any reference to such
I don't think such an intermediate state exists for very long. Once you shred the moon, it's shredded and the particles orbit on locally Keplerian orbits.
This would look like Saturns rings, which are also well inside the planet's Roche limit.
The intermediate state you propose to name would have to form either
a) a steaty state of in some dynamic variable or
b) have a sufficiently long lifetime (which would be your job to prove)
in order to earn its own name.
Edit: As OP has asked for material on ring spreading, I can point to A. Crida's work who had a bit of a controversial idea about the origin of the gas giant moons. However his calculations involve also classical calculations of collisional disc spreading.