what starts the pre-foreclouser criteria for how foreclosure information is published in ny state ?

Once a lender files a foreclosure case against u it is part of public record. This means anyone who wants to look at the filings or 'lis pendens' can find out that a foreclosure action has started. There r private companies who collect this information & sell it however the lender is not required to publish anything until a judgement of foreclosure is entered against u & a sale date is scheduled. The lender then has to publish the sale over a 6 week period in the local paper.
Preforeclosure typically refers to a homeowner who has had a foreclosure action filed against them but has a purchase contract to sell the home. Even if the purchase offer is for less than what is owed on the lender could consider agreeing to allow the seller to sell the home for the lesser amount to avoid foreclosure.

Pre-foreclosure is a period where the homeowner is being warned by the lender that the house is in default (is behind mortgage payments & unable to catch up) and need to do something about it, but at this point, the lender cannot claim back the property and sell it to recover the costs. But at this stage there are many options available for the homeowner to stop foreclosure like Loan Modification, Forbearance Agreement, Deed in Lieu, short sale etc.