The actual language from the NY Department of revenue website is given in the answer. The information you are quoting above is for NY nonresident filings only. (NY nonresident and part-year residents use the same NY state tax form but have different filing requirements). see this link for more information on part-year filings - <a rel="nofollow" target="_blank" href="
https://www.tax.ny.gov/pit/file/part_year_residents.htm">https://www.tax.ny.gov/pit/file/part_year_r...>.
So if you are actually a NY nonresident then you will not need to file an NY nonresident state income tax return if you do not meet these NY nonresident filing requirements.
