Take a pen and paper and write down exactly what you will say. Then call them back. This way you will make sure you say what needs to be said, you will not forget to say your name at the end and also share your phone number.
If your landline doesn’t have a way to save messages already built in, the best way to save that gorgeous message is with a recorder. But before you rush out to get it, don’t forget that most smartphones have a built in recorder. Try that first and see if it will pick up her voice loud and strong. It should save the message as an MP3 automatically, that you can send (via email or text) directly to your computer for safe keeping. If the quality is not great, then you can borrow a recorder or purchase one. We like the Olympus brand ourselves, because they save in MP3 and can upload to a computer wirelessly. We have the vtech phone/answering system indicated above. The outgoing message/greeting is of my young daughter’s voice and we would like to save it somehow. Am wondering if you have a process or can suggest how an outgoing/greeting message can be downloaded for safekeeping.

Here are some helpful links to help you record voicemail greeting that is clear and professional. Record your phrases for recording a great voicemail greeting here. Record yourself and compare it with a native speaker. Hear Georgie giving helpful feedback to other students on their voicemail greetings.
Website: https://shoretelcommunity.force.com/s/article/How-to-Manage-Your-Voicemail-Greetings-Using-a-Cisco-Phone