I have reviewed the Babylon code and something that may help you is the definition of a "rooming house" which you were cited under. The code states that a "Rooming House" is:
ROOMING HOUSE - A dwelling, other than a motel or hotel, where lodging is provided for compensation
If you are not taking compensation, then this would qualify. Here is the link to the code: http://www.tech.farmingdale.edu/~anderse/Code/BabylonC213Zoning1007.pdf
There is no other portion of the Babylon code that will assist you, however, there is a large body of case law in New York State on this issue. The NY Attorney General's office has summarized it well on this link:
Hopefully some of these cases will assist you in your situation
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