*One thing that you didn't actualy give is a defintion for "order of growth", which would explain why O(1)==O(2), and why O(2N+3) == O(N).*
From a casual perspective, "order of growth" can be thought of as the shape of the worst-case growth curve, regardless of the slope of that shape. Even it a line indicating linear (i.e., O(N)) growth has a high slope, eventually--when the data set is big enough--it will be overtaken by an O(N^{2}) curve. That's one of the pitfalls. People get seduced by the behavior of their code on small data sets, then get blindsided when they try a much larger data set and performance sucks.

Comment onRe: Re: An informal introduction to O(N) notation