Theoretical work on validity of restricting movement of Centroid of K-Mean
I highly recommend finding a source explains how k-means work and understand it well. The K-means is well known, so it is hard to find a reference talk about it as an algorithm or explain how it work.

I noticed you stating "author used ~1000 "fake" data points, so that the final centroid of K-mean stays within the required range" which is always going to be true. K-means is about calculating the mean (average) of data points used, which assure (always) to end with a centroid/s within the range of data used.

The power of this algorithm (K-means) is calculating the mean iteratively to reach stability of means (centroids). In another waord, in each iterate, means shift to be centered of denses. That give, if you in case of finding 1 K (one centroid) you will find it by one iterate.

Me personally suggest start with some videos, and go forward. Here is the first result on YouTube about k-means https://youtu.be/_aWzGGNrcic.