Another vote for YABU.
At the risk of repeating points already made :) :
Text messaging, like emailing or sending postcards, is asynchronous communication. You can send the message when it's convenient to you, and the recipient can read/deal with it when it's convenient to them. Messages are not guaranteed to be delivered in a timely fashion, in order, or indeed at all.
If you want to be alerted to incoming messages, you can choose to so be. Or not, if you don't. Your device. Your job to set it up how you want it to work.
Anyone trying to break the beautiful simplicity of this communications method by shifting the responsibility for notifications to the sender is being unreasonable.
If you really don't know how your phone works, I guess it's OK to ask the sender to modify their behaviour. But they are not required to do so, and the more insistent you are, the more likely they are to ignore your demands.
Unreasonable: "Please don't send me letters which will arrive on a Saturday as I like to lie in and the dog barks when the post arrives if we haven't bothered putting him in the kitchen overnight"
Unreasonable: "Dear BBC, please hold off from breaking any news stories after 9pm as it makes my iPad play the BBC News theme and it makes me jump"
Unreasonable: "Please don't @mention me on any posts you make on Facebook this afternoon because I'll be driving and I don't want the notification to distract me"