When you do something like:
wget -qO- http://example.com/path/to/script | sh
You're just running a remote script. This could install something or just run something. It can do pretty much anything because it's a script and scripting's what Ubuntu was built on.
When you use apt-get:
sudo apt-get install winetricks
You're installing it into /usr/bin for all users. The package you're installing is now coming from the Ubuntu repository which tends to mean somebody is looking after it and monitoring what code is going into it. Most people would consider a lot safer than running a random script off the internet.
Edit: In your case, you're just downloading a script - you're not running it inline like I was suggesting you might. What I've said still applies - you're just downloading it directly from whoever makes it but you're not installing it system-wide.
If you stick it in ~/bin/ (create that if it doesn't exist) your user will be able to call it from whatever path, otherwise you will need to prepend the path when you run it eg
Just to answer the question:
What do you call it when a program is registered with the OS from apt-get so you can just open it easily from terminal/dash?
- Installing a package describes that.
- There are scripts that could be downloaded and run (in one movement, like my first example) which would also be counted as installing, but it's not a managed package install.
- If you're just downloading a script with wget, I'd have to call that "downloading".