In this particular example, it is "the housing authority" of "New York City". There is, presumably, only one housing authority there. If that's not correct, then the definite article is inappropriate.
I believe that, when I gave the information you posted, I was referring to things like "London Bridge" or similar. There is more than one bridge in London so it's not appropriate to use the article.
In any case, you should know by now that very few of our "rules" are hard and fast. There are exceptions to every rule and there is plenty of quirkiness in the English language.
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