Edward T: I would like to know where you are reading this from? Osha states in 29CFR 1910.178:

1910.178(l)(1)(i)
The employer shall ensure that each powered industrial truck operator is competent to operate a powered industrial truck safely, as demonstrated by the successful completion of the training and evaluation specified in this paragraph (l).
1910.178(l)(1)(ii)
Prior to permitting an employee to operate a powered industrial truck (except for training purposes), the employer shall ensure that each operator has successfully completed the training required by this paragraph (l), except as permitted by paragraph (l)(5).

1910.178(l)(2)(iii)
All operator training and evaluation shall be conducted by persons who have the knowledge, training, and experience to train powered industrial truck operators and evaluate their competence.
1910.178(l)(3)

No where in the Standard can I find that it is the Instructor's responsibility to maintain these records. Only that the Employer is responsible to make sure that their employees are trained under the standard. I've been in the Business for 25 yrs and Training for most of it.