From the best of my knowledge this case requires "there" and accepting "in this letter is an invitation" was a mistake. It's grammatically incorrect. Perhaps this is one of the common mistakes of native speakers on spoken English, but incorrect, specially on writing. However feel free to prove me wrong
Well, yes, the words are different; but I fail to see how the meanings differ.
If I were translating from Russian, "This letter contains an invitation" would be the first translation to pop to my mind -- that's what actually happened when I was working on this lesson.
On the contrary, literal equivalent of "This letter contains an invitation" in Russian sounds pretentious and rather unnatural for everyday speech, in Russian I would say "There is an invitation in this letter".