Wow, that sounds like a very poor job training for the support guy.
It is very simple.
Every authentication procedure is basically between you and something else. If you keep that "something else" in mind, then you know (hopefully) what to do. It is like trying to use your username/password of you gym membership on an ATM machine. It doesn't work.
Apple ID
You use that username/password for everything related to your Apple Account and whatever is linked to that account: Your App Store purchase, iCloud, iMessage, Mail, etc.
OSX User
This has nothing to do with any business "outside of your computer". It is your key to your data on your computer. Something is written/overwritten to a sensitive directory on your computer (i.e.during an app installation), you need to verify that you have access to that directory/file on your computer.
Confusion:
Here is an example where the two worlds overlap: When you purchase GarageBand X from the App Store, you have to authenticate with your Apple ID to do the purchase. When continuing through the download/installation process, there comes a step where the installer wants to save the sound files to your system directory. At that point you have to authentication with your OSX Admin user name to grant the access to your computer drive. The confusing part is that many users still think that this is part of the purchase procedure requiring the Apple ID.
Education
The username/password scenario is a perfect example why I spend so much time for details and extra illustrations in my manuals. The download/installation procedure is in many books not explained at all or you find any YouTube videos for that (installation is not such a cool topic) and now you know from your own experience how much time, energy, and frustration that can lead to.
Here is an excerpt from my book "GarageBand X - How it Works", where I explain on several pages the step-by-step producer of the entire installation process of GarageBand.

Hope that helps
Edgar Rothermich - LogicProGEM.com
(Author of the "Graphically Enhanced Manuals")
http://DingDingMusic.com/Manuals/
'I may receive some form of compensation, financial or otherwise, from my recommendation or link.'