If you have been searching "DUI lawyer" and seeing "DWI" everywhere, you are not confused, the terms just vary by state. Here is the plain-English version for New York.
New York uses "DWI," not "DUI"
"DUI" (Driving Under the Influence) is the common term in many states, and people everywhere use it. New York's statute, Vehicle & Traffic Law §1192, does not use "DUI" at all, it uses DWI (Driving While Intoxicated) and DWAI (Driving While Ability Impaired). So when a New Yorker says "DUI," they almost always mean a DWI. Same situation, different label.
New York's actual charges, low to high
DWAI-Alcohol, BAC more than .05 but under .08, or observed impairment. A traffic violation on a first offense, not a crime.
DWI, BAC .08 or higher, or proof of intoxication. A misdemeanor on a first offense.
Aggravated DWI, BAC .18 or higher. A more serious misdemeanor on a first offense.
DWAI-Drugs and DWAI-Combination, impairment by a drug, or drugs and alcohol together.
Zero Tolerance, drivers under 21 at .02–.07, handled through the DMV.
Why the label doesn't change the defense
Whatever you call it, the defense is the same discipline: was the stop and arrest lawful, was the testing reliable, and can the charge be dismissed or reduced to DWAI? Jonathan handles all of these, DWI, DUI, drunk driving, drugged driving, for a flat $1,500 misdemeanor fee.