You are quoting from previous edition of patent office manual. The current version of Manual of Patent Office Practice and Procedure (2011) has no reference to the text you have quoted. Please note that manual is only for guidance. The patent act and rules remain dominant if there is any doubt or inconsistency. There are no such things in the act & rules.
Even if I take the above sentences to be an existing practice, you seem to have ignored the dominant text in the paragraph. It is very clear from the first sentence that the onus is on the patentee to ensure that the renewal fee is timely paid and the patentee cannot ordinarily escape this responsibility by showing his innocence. Proving negligence on the part of the patent agent may be an exception to protect the legitimate interest of the patentee, but not a rule.
Personally speaking, I will not restore the patent if the agent was negligent in renewing the same. This is because the patentee has given an undertaking that he is bound by the acts of his agent. So it is his responsibility to ensure that his agent acts in a timely manner. Please note that a patent agent may be disqualified on account of misconduct. It would be a different debate if negligence equals to misconduct, but it is certainly a highly unprofessional conduct.
As to your other question “what kind of evidence should be produced”. This is fact specific; you are the best judge for the circumstances you are in. For example, if a Japanese applicant failed to renew his patent he may be allowed to restore his patent if he is able to establish that his place of business was located in one of those cities affected by earthquake and tsunami. For example, USPTO offered certain flexibilities in meeting deadlines to Japanese applicants who were affected by earthquake and tsunami.