What I don't understand is why the OP and her mom don't simply discuss this for 2 minutes and figure out the best way to communicate. Why go for unhealthy indirect manipulation of behavior when it's so easy just to discuss it directly? Of course, that assumes that what they really want is a solution to a little problem, not a victory of a power struggle.
It sounds so foolish to me at both ends: Mom knows OP doesn't like voice mail, but she insists on only communicating that way, as though this will cause OP to suddenly start checking her voice mail often. OP knows that this is the only way Mom will communicate, but she refuses to listen to the messages, as if Mom will then send a text -- and she doesn't change her outgoing message to state that she rarely checks voice mails.
We have showed our moms how to use the text messaging features on their smart phones. It's so handy when we are sending a message to the whole family, especially when we're traveling. But we also know that they don't like it and they simply don't think in terms of text messaging, so if we want to be sure that they read what we're sending, we send the info in an e-mail or call, too. It takes just a few seconds more -- certainly less time than a failure to communicate at all would often waste.
With all due respect, I think that the OP and Mom are both being a little stubborn, a little PA, and a little silly, frankly. It's not the biggest deal in the world, but it sounds like they both prefer to make this a problem and be the one who's right than to figure out how to communicate effectively. And it can easily affect others, too, depending on what the message is about, as it even kind of did here when it led to OP missing a family event because they played this rather childish game.
I think the nicest thing would be for each to consider the other's preference and put the other person first, as we do in so many areas. Mom, text OP. OP, call Mom. But if the other person doesn't do it your preferred way, roll your eyes and listen to your message/read your text. And may this be the biggest problem you ever have.