Is this true? Do I need to install WordPress only once, even though I am setting up two websites?

NO – You will be using the exact same database – you need to use a separate database for each installation. If you use the same database, this error will show.

Given the fact that I installed www.ads-software.com onto my computer while setting up my own website and hosting it on APlus, does that mean WordPress is now truly on my computer, as I was told, or is it somewhere else, such as at the host company’s server?

WordPress can be installed anywhere- it is most likely on both your computer, ‘APlus’ and ‘host company’s server’ – all separate installs and instances of WordPress.

If, and when, I finally locate WordPress that I installed, where do I look, to see which version (the ‘.org’ or the ‘.com’ version) got installed?

You can’t install .com version anywhere but .com. WordPress is WordPress – when installed on .com, it auto-installs a locked down version. .org allows you to download and install the full version yourself.

Assuming that I can actually find WordPress, can it be used to set up my son’s website, even though I have already used it to set up my own website?

Yes, you can install WordPress as many times as you like in as many places as you like as long as you have:
– unique URL for each install
– unique database for each install

If I eventually get both websites up and running, am I correct in assuming that I can then add/delete/change things on either website, no matter where I am or what computer I am using, just so long as I can log in to each site, using the administrator’s user name and password?

Only if you can access that install from the location you are at.
e.g. if you install on your laptop and your laptop is OFF, obviously you can’t access the site from another computer.

Basics:
You need a domain name.
You need to point the domain name to a server/computer connected to the internet (using DNS)
You need to install WordPress on the server/computer using that domain name.
You need to provide that WordPress installation with it’s own database to use

Then you just access WordPress any time via the domain name.

https://www.webhostingsecretrevealed.net/web-hosting-beginner-guide/