I’m a retired realtor and VP of my co-op Board. When buying a co-op in the NYC area, you do need to fill out a UCC-1 and pay the filing fee. You do not have anything to do with a title insurance company and nothing to do with lender’s insurance - period. In NYC, most co-op Boards require a minimum down payment between 20 - 30% so there is no need for lenders to ask buyers to pay lenders insurance. By the way, in the nearby suburbs, typical co-op Board down payments are 10 - 20% (much cheaper than the city).
Since you’re buying a co-op, make sure to have your attorney or lender review the financial documents of the co-op BEFORE you get to closing. You’re entitled to review them too. There are plenty of co-op and condo buildings who mismanage their financials and are forced to increase monthly fees frequently, or they have pending lawsuits, or some other issues. Reviewing the financials is critical because if the Board can’t run the building efficiently, then you’re stuck with the mess after you move in.