My comment in regard to training notes was more in regard to solo flights and after they complete training. During dual training the instructor is PIC, so carries the full burden of responsibility. If tgat instructor works for an entity such as a flying school, and complies with that schools policy and procedures then the financial liability will rest with the employer.
On a side note, as a joke. I remember an episode of Judge Judy where a student flying nav or area solo tried to sue the instructor for a forced landing following engine failure, due no fuel. The instructor provided evidence that the student had been signed off to daily and fuel the aircraft themselves, so Judy threw out the case noting that the instructor must be competant as the student sucessfully conducted the forced landing without power as trained, without injuring themselves.