I can leave messages on my AT&T land phone, but the phone does not show that they are there

*298 is used by COX cable to retrieve voicemail messages from your home phone, you left that out. The message light on your phone or any phone with a voicemail message indicator is controlled by your provider, not by the phone. You'll have to report this to your provider, it's not a device issue. If you have further questions provide the model number of the phone you're talking about. 

Your first sentence is correct, but so what? Why should that stop your phone from doing its job?

Read it again. Your provider isn't sending the code that would turn the phones indicator on. It's NOT the phone. 

I just called Cox. They told me that they would never override the system that the phone uses. They did say that different phones use different systems but they wouldn't interfere.

Then replace your phone. The new one will do the same thing. This is a common problem with cable companies. They can't even spell FSK code. 

Just to clarify one thing, your messages are being left on your phone providers voicemail, not your phones answering machine. So, you're not leaving them on the phone, like you have stated. 

Thanks, but I have written to your Home Office for an explanation/apology.

Sincerely,

Warren Jones

Thanks, but I have written to your Home Office for an explanation/apology.

Whose home office?  Apology for what?  You've gotten a very good explanation.  The "Answering Machine" you are using is provided by Cox Cable, your telephony provider.  The message waiting indicator on your phone would be controlled by your telephony provider.  It sends a signal to turn it on, it sends a signal to turn it off.  Your phone has no idea about what messages are available at your telephony provider outside what it tells it.

Any apology needs to come from, Cox Cable, or you.

Thanks, but I have written to your Home Office for an explanation/apology.

My home office is my home. I've been retired for some time. I don't think you realize this is a customer-based forum, you aren't communicating with AT&T. Did you try a different phone? Did it do the same thing? Having been in the "business" for quite a while I do know how the network works.