I would have to say that they are an ACH originator if your bank is originating the ACH transaction. In other words, if they just have reoccuring ACH transactions on their account but another bank is initiating the ACH transaction (i.e., DISH payments) then no they are not an ACH Originator.
In our bank we have one method for ACH origination and that is through our online banking system. We have to grant them access to this (after agreements are signed) and we have all kinds of controls in place and operational procedures.
If you have two methods for your business customers to send you ACH files for processing then you should have two sets of procedures for this.