The definition of "complaint" isn't specific to an employmer or even whether or not the complaint comes through your regulator. Your policy need not differentiate between a consumers letter and one from a regulator.
Some regulations place time frames and requirements upon you no matter where the complaint originates. Regs E, Z, FCRA and RESPA in particular.
I recommend your policy look at the problem and resolution, that is what is important. Take a broad view.
"(b) Consumer complaint means an allegation by or on behalf of an individual, group of individuals, or other entity that a particular act or practice of a state member bank is unfair or deceptive, or in violation of a regulation issued by the Board pursuant to a federal statute, or in violation of any other act or regulation under which the bank must operate."
Opinions stated are not necessarily that of my employer.