Richard: I had forgot myself. Am I not king? Awake, thou coward Majesty, thou sleepest!Is not the king's name twenty thousand names? (3.2.5)
The moment Richard realizes just how much danger he's really in, he tries to rally by remembering that he's in a position of power. He's king, after all – a powerful name. Of course, he's right in one sense: the king's name is twenty thousand names, in the sense that the office of the king commands the loyalty of his twenty thousand or so subjects. What Richard fails to realize, though, is that he might not in any real sense be king anymore, because Henry Bolingbroke is about two seconds from forcing him to give up the crown.
Richard: Arm, arm, my name! (3.2.5)
This is kind of a weird moment, don't you think? Here Richard asks his own name to take up arms on his behalf. In other words, he thinks that language – both the title of king and his orders as king – becomes almost magic because he's king. There's something spell-like about this moment when Richard asks his own identity to defend him.
King Bolingbroke... (3.3.4)
When Henry Bolingbroke shows up with an army and backs Richard into a corner at Flint Castle, Richard knows he's not going to be king of England much longer. Here, Richard's sarcastic address to Henry does a lot of work. By refusing to call him "Lancaster," the title he should have inherited when Gaunt died, Richard is indirectly defending his seizure of Gaunt's property. Also, because "Bolingbroke" is a title that specifically refers to a duke, by calling him "King Bolingbroke" Richard is trying to show how ridiculous it is for Henry Bolingbroke to even try to assume an identity that can't, by definition, be rightfully his.