I think I agree with you to some point. There have been a couple of suggestions and it's unfair of me to attribute them to you.
My concern is with a physician's office (or a bank, etc.) calling from a phone that does not provide Caller ID information and leaving a message that goes something like: "This is a message for you, please call us back at your earliest convenience." This would result in a call that I would ignore, followed by a voicemail that I would also ignore. Effectively it would mean that there would be no phone communication between myself and this office, bank, etc.
In my opinion, when you provide a bank or physician's office with your cell phone number and authorize them to contact you via that method, you are agreeing to let them leave the minimum required information. In my opinion, I need to know (1) who is leaving the message and (2) is it important. In a pinch, I will settle for (1). Without either, it may as well be static on the phone for all of the information it conveys. I disagree that I am allowing these offices to treat my information "less sensitively". On the contrary, it's my responsibility to treat my end of the communication channel sensitively.
A code word is an interesting idea, but people are pretty poor at remembering arbitrary information (on the whole). It's common for many people to use the same password for every website, since they have trouble memorizing more than a couple at a time. Code words, I fear, would have the same issue. People who don't need a code workd would forget it and they'd just be confused with the message; I think it would need to be opt-in for those who cannot secure their own side of the communication channel (voicemail, email, etc.)
Some of the electronic medical record (EMR) systems that have a public facing web interface do provide a more secure method of communicating with clinician or their office. In my experience these often mimic email and are secured via SSL, they require the typical username and password (or PIN) combination. These will probably become more popular, but I expect clinicians will still fall back to phone calls if the information is time sensitive.
For those who insist on listening to voicemail via speakerphone, it seems like the smart move is to ask the physician to contact them only through a secure website or email and never via cell phone.