"As a result the 250 W is about right for most of the earth's surface - not including clouds."

Well, they said weather and weather includes clouds- by [at least I would imagine] most people understanding of the word "weather". Though btw, I've heard it said that light clouds do not block a lot of energy as far as solar panels are concerned.

The quote:

"... but by the time it reaches the ground, it has been reduced by atmospheric absorption and scattering; weather; and summer, winter, and day‐night cycles to less than an average of 250 watts per square meter."

Going back:

"The atmosphere gobbles up a lot of energy before it hits the ground - so much so that even with rotating collectors the power coming through is quite low at early morning/late afternoon because of the extra path-length of atmosphere the light passes through."

Well if 1000 watts reaches the ground and before the atmosphere there was over 1300 watts- that's around 30% of energy- and you could call that "a lot of energy".

I don't have any number handy, but I doubt that more than 50% of energy is lost because sun is low on horizon. So in other words if it clear and without much haze, I would expect you would get at around say 800 watts just prior to Sun beginning to sink below horizon. Of course if Sun is halfway or more down the horizon, the earth itself would be directly obstructing the energy.
But if you got some numbers I would like to see them.