bluewave you seem like a bright guy so offering this up more so you know in the future. 'climate normals' is a term of art and is different from 'normal'. 'climate normal' is just a synonym for "average" or "arithmetic mean", but applied to a specific period of time. Generally in climatology when we speak of "normal" in the sense you're using it, we mean conditions that are within 1STDEV of the mean (most climate statistics in most locations are roughly normally distributed, so this conceptually works). In other words, it is common for NY to have below average snow, or above average snow, but still within the "normal" range. A year like 1997-1998, on the other hand, was both below average and not normal.
I am aware meterologists use below normal as a shorthand for below average b/c of the NWS terminology and to make it easier to communicate to the public, but it's not a rigorous way to describe it and tends to blur the difference between events that are extraordinary and events that are typical.