For Ohio: The NOF for Private Construction Project appears to "only" require a statement: Please take notice that undersigned is performing certain labor or work or furnishing certain materials to while an NOF for a Public Improvement Project requires a "description" of labor, work, and materials.
You are correct. The notice of furnishing on a private project does not need a description of the work (at least that is how the statute reads). There is also no relevant case law (that I know of) on the subject.
The way I instruct clients to think about it is to look at the purpose behind the statute. In Ohio, a NOF only must be served after the owner first records a notice of commencement (NOC). The NOC serves as the owner's notice to the world that a construction project is commencing on the owner's property, and it provides notice to subcontractors and suppliers that if they wish to keep their lien rights intact, they must serve an NOF.
So the purpose of an NOF, from the subcontractor or supplier's perspective, is simply to provide a notice, to the owner, that the sub or supplier is doing something related to the project. A sub or supplier doesn't need to provide the NOF if it is under contract directly with the owner, so the purpose of the NOF is for the owner to keep track of the contractors working beneath the general contractor, so that the owner knows who it needs lien waivers from, etc. before making payment to the general contractor.
Thinking about the notice as just that, a notice, leads me to believe that courts would be less concerned about the description of the work than the fact that the notice was served properly to the owner, giving the owner an opportunity to perform its due diligence and determine who the sub or supplier is, and what the sub or supplier is doing on the project.