Anderson's "The Man Who Came Early" is a first person narration addressed to a single auditor, a priest. "The Light" is another first person narration addressed to a single auditor, this time a historian who is being told government secrets - and the surprise ending of the story reveals why. Someone reached the Moon before the Americans. (The background of da Vinci's The Virgin of the Rocks does look as if it might be a Lunar landscape, doesn't it?)
"The Light" is a first Moon landing story published in 1957, so how much did it get right? Americans, three of them, previous circum-Lunar missions, an embarrassing "first man on the Moon" speech written in advance by someone else. However:
they set off from a space station, not from the Earth's surface;
all three go down to the Lunar surface;
they have no radio or television link back to the Earth.
Thus, the first to step onto the surface can get away with:
"'May I suggest that the captain write in the log that the speech was delivered?'" (p. 170)
Description of the Lunar surface: eerie; bright; dead; huge, razor-cornered rocks; horizon near; deep, sharp shadows; cracked, ocherous land; indescribable light; a lava plain like polished black metal; oven in sunlight, freezing in shade; a mysterious mist; meteoric dust; a familiar fog-glow; prints of hob-nailed boots worn by a tall man...
The First Man on the Moon learns the identity of the real First Man on the Moon.
In a pre-Apollo cinema adaptation of The First Men In The Moon, a UN Moon expedition discovers that Cavor and Bedford were there before them. A post-Apollo TV adaptation asks us to accept that Wells' account of Cavor and Bedford and the TV transmissions from Armstrong and Aldrin both happened, then ingeniously fits both of these "first Moon landings" into a single narrative.