The slant of that website is, uhhhhh,
impressive, but let's set that aside and go to the primary source[1]. First, it's a self-reported survey, not a study. Tellingly, it doesn't list any absolute numbers, just "yes/no" reporting. For example, 77% of businesses reduced hours hours per-employee, but it doesn't say by how much. That doesn't seem at odds with the study being discussed in this thread. That is, hours may be reduced, but overall wages and employment remain up relative to other industries. Actually that's another interesting thing about the survey you link. It cites a 6% to 1% drop in growth rate, but doesn't compare that to other industries, as the study in this thread does. The survey also says that food costs went up (an expected result from the wage increase), but they didn't ask what the impact on revenue was.
The survey you cite seems low quality anecdata compared to actual industry numbers shown in the study being discussed.
As to your first point, no, "the minimum wage hike" did not occur in late 2018. The final increase ($13.50 to $15.00) did indeed take place near the end of this study period, but the overall increase from 2013 to 2018 (from $7.25 in 2013 to $13.50 by late 2018 to $15.00 by end of 2018) reflects a more than doubling of the minimum wage over the study period, so I think it is still a valid reflection of the impact of such a dramatic increase.
[1] https://thenycalliance.org/assets/documents/informationitems...