Not the specifics, unfortunately. Much of what we know comes from Unfinished Tales:
He has a few other names that aren't mentioned in the question, but most have similarly vague origins:
Greyhame is used exclusively in Rohan. As noted by Wayne Hammond and Christina Scull in A Reader's Companion, this is a modernization of the Old English græghama, which means "grey-coated." This name probably dates back to Gandalf's first appearances in Rohan, since it's the obvious name for him
Stormcrow is used mainly in Rohan, but also by Gandalf himself at the borders of Gondor. Although we don't know where this name came from, or who coined it, it's likely that it came about because of the perception of Gandalf as a bearer of bad news (as with Láthspell) below; as one of the Gondorian guards says:
'May you bring good counsel to Denethor in his need, and to us all, Mithrandir!' Ingold cried. 'But you come with tidings of grief and danger, as is your wont, they say.'
Return of the King Book V Chapter 1: "Minas Tirith"
In only two cases do we know both the specific individual and the circumstances surrounding the name:
Láthspell was probably given by Wormtongue; it's the first we hear of the name, anyway:
Why indeed should we welcome you, Master Stormcrow? Láthspell I name you, Ill-news; and ill news is an ill guest they say.' [Wormtongue] laughed grimly, as he lifted his heavy lids for a moment and gazed on the strangers with dark eyes.
The Two Towers Book III Chapter 6: "The King of the Golden Hall"
The White Rider was likely coined by Aragorn:
'Do I not say truly, Gandalf,' said Aragorn at last, 'that you could go whithersoever you wished quicker than I? And this I also say: you are our captain and our banner. The Dark Lord has Nine. But we have One, mightier than they: the White Rider. He has passed through the fire and the abyss, and they shall fear him. We will go where he leads.'
The Two Towers Book III Chapter 5: "The White Rider"
It's later used by Gandalf himself, the Eorlingas (who learn it from Aragorn), and Pippin (who likely heard it from Aragorn, Legolas, or Gimli after the Battle of Helm's Deep)
Unfortunately, that's the most we know about how these names came to be applied to Gandalf. The specific circumstances, and the first individuals to use each name, is unrecorded.