If the original IRA custodian sends you a check for you to carry to the new custodian and the check is made out to the new custodian FBO the your IRA, the movement of the funds still constitutes a trustee-to-trustee transfer, not a distribution and not a rollover.
A trustee-to-trustee transfer is not subject to the 60-day rollover deadline (because it is not a rollover), but a delaying completing a trustee-to-trustee transfer that long makes no sense since you can't do anything else with the funds.
