The trustee/custodian of an IRA is the entity that holds the IRA. A 403(b) is not an IRA, so the movement of funds from an IRA to the 403(b) is not a trustee-to-trustee transfer of one IRA to another. Any movement of funds from an IRA to a 403(k) *must* be reported on a Form 1099-R.
If the bank refuses to provide you with the 2014 Form 1099-R that was required to report the rollover from the IRA to the 403(b) and the IRS questions the rollover, you'll need to provide explanation to the IRS. The amount distributed from the traditional IRA and rolled over to 403(b) should have been included on your 2014 Form 1040 line 15a or Form 1040A line 11a but excluded from line 15b or 11b and the word ROLLOVER should have been present on the line. Since it would have been excluded from line 15b or 11b, there is no tax liability for the rollover and so there is probably no need to amend your 2014 tax return as long as you can provide explanation if the IRS ever questions the rollover.
