You are comparing 2 different things. The first link is about Windows Azure Storage
while the second link talks about Windows Azure Storage Analytics. I'm assuming both links are correct for the part they're representing.
Storage analytics data provides information on the storage requests. And the explanation for ThrottlingError is given as "Authenticated request that returned an HTTP 503 status code". When requests are throttled then also 503 status code is returned, right?
I think both links talk about the same thing