When asking "date established residency there?" Do they mean the date I was officially considered a legal resident based on state laws or Do they mean when I moved there?

When you started living there.

New York determines your residency for tax purposes if you are domiciled in New York. They define domicile as: your permanent and primary residence that you intend to return to and/or remain in after being away

This link will tell you more:

NY Residencey

When you moved there.  You're misinterpreting the 180 day rule.