To set up your phone, go here. You put in your number, then I believe (it's been a while since I did it), they send a text to your phone to confirm, giving you the number with which you will text in your updates.
You don't need to 'phone' any password in. It's a question of verifying the number matching your account.
No, I think that's for people who are already using a mobile to update Twitter.
To set up a phone to use it to update Twitter, you go to 'Setting' (top right). Then 'Devices' (top). Then, once you enter your phone number, it tells you to "Verify your number +17971xxxxxx by sending the following code to 40404:" Then it give me a random code.
But I have no signal here, so I can't send the code.
I should have tried it today while I was out, but I forgot.
