Re: Education...but by who?
"You can prove that you are the bank by getting them to phone you back on your bank number. It's pretty certain that if you phone a bank, the person you end up speaking to works for the bank."
There's actually a well-known scam based on this. The scammer puts a recording of a dial tone on the line to fake having hung up and then an accomplice takes over the call after the mark has gone through the motions of calling back. Ring-back verification only works if you call back on a different line.
"There really is no answer to this one, as eventually one side has to trust the other, but the banks are aware of this"
They show no indication of such awareness. The proof that they expect you to offer is the sort of information a scammer would need to impersonate you. It's no different to a faux website at www.somebnk.com collecting passwords from a mistyped URL. The onus has to be on the originator of the call.
As to your challenge there are three responses to this
(a) it's the banks' problem - they should have solved it before they started making the calls;
(b) you have an agreed set of information which they will use to identify themselves and which is different to that which you use to identify yourself, a solution so blindingly obvious that even the USPO should be able to reject a patent application;
(c) absent any such arrangement a few moment's thought should reveal to you an obvious technique which you can apply unilaterally, which works equally well with passwords if you're not sure the site is genuine and which will actually impede the fraudster so long as everybody doesn't start using it.