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Not many people realize that the literature you mention is the field of "analysis of algorithms", which is a sub-field of (or, in practice, somewhat different from) computational complexity theory / theory of algorithms. Robert Sedgewick (CS professor at Princeton, and an early PhD student of Knuth) has a great book with Flajolet on Analysis of Algorithms , and in one of the lectures from his course  makes a distinction between the complexity analysis usually taught in basic undergraduate algorithms courses (he calls O-notation not the scientific method, in a certain context in the lecture) and AofA (which involves saying "Running time is ~aN^c" instead of saying "Running time is O(N^c)", and also actually measuring against real programs) — watch the video or read the slides; it's an interesting distinction.
And the Purdue website  is even better at giving a sense of the field.
Oct 29, 2016 · bogomipz on Big-O notation explained by a self-taught programmer
Its both. Here are some links:
Video Lectures are here: https://www.coursera.org/learn/analysis-of-algorithms
and the book:
Also a youtube - short history of algorithm analysis by Sedgewick is worth a watch: