Why did Théoden suddenly trust Gandalf?

[Alcuin]

In Two Towers, “King of the Golden Hall”, Théoden greets Gandalf gruffly, calling him “Stormcrow”:

[W]hen I heard that Shadowfax had come back riderless, I rejoiced at the return of the horse, but still more at the lack of the rider; and when Éomer brought the tidings that you had gone at last to your long home, I did not mourn.

Wormtongue then adds his derision, insulting Galadriel almost as an afterthought, calling her “the Sorceress of the Golden Wood,” saying that “webs of deceit were ever woven in Dwimordene.”

Gandalf responds with what seems to be an old rhyme of Rohan concerning Lórien and Galadriel. Then he casts off his old, grey cloak, revealing himself as a shining figure in the darkness. (A storm is passing over Meduseld at the same time: they soon see it moving westward when Théoden steps outside.) He declares, “I have not passed through fire and death to bandy crooked words with a serving-man till the lightning falls.” And it seems lightning does fall – from Gandalf (though he normally seems to use fire rather than lightning) or from the passing storm from the East – upon Wormtongue, who “sprawled on his face.”

Immediately Théoden rises at Gandalf’s invitation and steps outside.

Why does Théoden’s do this? Gandalf just struck down his counselor, and whatever the rest of the court at Meduseld thought of Wormtongue, Théoden trusted him. Is he going with Gandalf out of fear? is it awe (related to fear, though different) because of the display? has Wormtongue’s mask* been torn partly away so that Théoden can see him in part as others do?

Why does Théoden suddenly rise and go outside with Gandalf, listen to him on the steps of Meduseld, and in a trice change his whole demeanor, intention, and direction?

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* I mean this as a symbol, of course: Wormtongue wasn’t wearing a physical “mask” separate from his face, nor did Gandalf injure the flesh of his face; but he presented to Théoden a “false face”, masking his true intentions. GrÃ*ma is Anglo-Saxon for “mask”, Gálmód for “lecherous”: Wormtongue is GrÃ*ma son of Gálmód, “lecherous mask”.