Is "imperative computing" just a fancy way of saying "scripting"? Or is there more to it?
Your understanding is not correct. There are two major programming paradigms: imperative programming and declarative programming.

In imperative programming you basically tell your computer what to do to solve a certain problem by providing an algorithm which might be implemented as a composition of different functions and/or methods. Often one distinguishes between object-oriented and procedural programming as the two main subparadigms of imperative programming, where the latter one is often scripting but the lines between scripting and "software engineering" (i.e. writing fully fledged programs) is blurry. Procedural programming can be engineerish, too.

In declarative programming you do not tell your computer how to solve the problem but rather what is the problem. Again, one might distinguish between subparadigms where one might be more frequent for hacky scripts where others are more profound.

You might want to check out C, Java, and Haskell (for instance) as procedural, object-oriented (i.e. the two mainly imperative languages) and a functional (i.e. a declarative language) language. Maybe implement functions to sort a list of integers in each of them, the differences between the latter as opposed to the first two should become very clear, but also the corresponding Wikipedia pages should provide good insights. However, e.g. in C++ (object-oriented) there are some features whose origins are in functional (declarative programming), e.g. lambda functions. On the other hand Haskell also has some sort of loops which are strictly speaking a rather imperative concept, so programming language designers adapt concepts from other languages from another paradigm, the line is blurred and not as clear as 30 years ago.



The difference between scripting and software engineering is even less clear. I would say it depends on the size of the program, reliability, use cases, how well it is documented, ... whether a certain program is a script or "fully fledged".