Gleims' information is correct, in part. However, a photocopy of your former certificate is not adequate to satisfy the requirements of the administrator. The reference to qualify this discussion will be found in the Federal Register Preambles found in Volume 62, Number 65, dated April 4, 1997:
"FAA Response: Based on public comments that argue the proposed section could create difficulties in situations where flight instructor certificates are mailed in upon completion of a renewal course, the FAA has decided to add the phrase "or other documentation acceptable to the Administrator," which would permit a flight instructor to use a copy of a graduation certificate from a CFI refresher course and a copy of the completed application for renewal to meet this requirement. "
In Volume 62, Number 146, on July 30 of that same year, the FAA further clarified that a copy of the application is not necessary to meet the requirement for "other documentation acceptable to the administrator," as specified in 14 CFR 61.3(d)(1):
"The FAA notes that the preamble to the final rule states that under paragraph (d) the phrase "other documentation acceptable to the Administrator" would permit a flight instructor to use a copy of a graduation certificate from a CFI refresher course and a copy of the completed application for renewal to meet the requirements of that paragraph. However, the FAA has determined that the latter document is not necessary. Therefore, a copy of a graduation certificate from a CFI refresher course, without the application for renewal, is acceptable documentation for the purpose of meeting the requirements of paragraph (d)."
AC 61-65D further states that " during the renewal process, when an FIRC graduation certificate is used for renewal and the applicant's flight instructor certificate is being processed for renewal, a copy of the FIRC graduation certificate is considered acceptable documentation by the Administrator until the applicant receives his/her permanent flight instructor certificate. Another example of ". . . or other documentation acceptable to the Administrator . . ." [per § 61.3(d)], is a copy of an applicant's completed and signed FAA Form 8710-1 that was submitted when making application for renewal."
Note that all of this must be accomplished prior to expiration of a flight instructor certificate. Once the certificate has expired, a practical test is required for reinstatement. AC 61-65D states "a practical test is required in the exchange of an expired flight instructor certificate. A designated pilot examiner or an ASI shall not reinstate expired flight instructor certificates without a practical test."
I would certainly never take an action that affected my certification strictly on the basis of a commercial company such as Gliem or Jeppesen. It's far better to know where you stand based on the regulation, than what "gleim said."