First, again for what seems like the millionth time, the phrase "Clean install", as is used in the Apple/Mac world, means doing just a clean, fresh installation of the Mac OS. Obtaining the needed files.folders, apps, etc. is done by the migration process which Apple designed to make it easier and smooth. Of course, the cleaner one's system is, the better. One can of course after the installation of the Mac OS do a one-by-one installation of any third party applications, but then what about settings, files, folders, etc. that are needed? That's why the migration process makes that smooth and reliable.
Whenever a new Mac OS, or a newer version of a Mac OS, is released, Apple offers it in two "flavors": either the full installer, or just an updater file. If one just uses the updater file, it will update the version of the Mac OS currently on one's machine to the new one. That is all that is done. No further migration/copying of files, folders, etc. is needed.
So, the steps involved in applying the updater file are dependent on whether one has the update done by Apple (ie, in the background), or applies it on their own. In either case, the first step is always the same: make a backup. That goes without saying.
Then, one can either have the update done automatically in the background, or download the updater file itself and apply it at one's leisure. Having it done automatically (ie, in the background) does rely on a decent internet connection, in terms of speed and smoothness. The advantages of downloading the updater file and applying it on one's leisure are 1) the internet need is just for downloading the file, and 2) one has a copy of the updater file. That is beneficial if it is necessary to apply the update again (for example, in a recovery situation).