Franc -- most brokers required it in writing. A few, Etrade being the notable one, allowed it over the phone, but to be followed up in writing.
In theory, the broker is responsible for keeping the files for a period of time, if there were any challenges (or proof) needed to how a vote was done.
While your phone-vote was certainly "on a recorded line" -- you can see how hard it would be to review thousands of those if "someone" wanted proof of how thousands voted.
Paper/fax copies are easier for that purpose.
As Clawmann suggested, it's probably a question for you to your broker on the specifics of their requirements.
Either way, if your votes are recorded and shares locked, you should be ok.
...Catz