Hello, I’m Nathan, and I’ll be assisting you today. My livelihood depends on your positive feedback, so your satisfaction is my highest priority.
Sorry to hear you’re having trouble.
What is happening is that your callers are reaching a voicemail service provided by the phone company (presumably comcast?). It is answering before your home answering machine has a chance to pick up the line, which is why you hear a different greeting, and why you get no messages when checking your home answering machine.
The solution in that case is to contact the phone company and have them cancel/remove the voicemail service from your account, OR as an alternative, you could get rid (or just stop using) the answering machine and use the voicemail function instead. You can record your own greeting on the voicemail if you like, and voicemail actually has several advantages: it is always on, meaning it will work even if the power goes out, while the answering machine could stop taking or playing messages if it is ever damaged or unplugged. VM will also work even if you're on another call, with answering machine, the 2nd caller would just get a busy signal. For these reasons, I tend to recommend voicemail over using your own machine, simply because it tends to be more reliable and accessible. The choice is yours of course though. If keeping VM instead, you'd want to have them increase the rings, 4 is the standard typically.