As is usual in stupid arguments on the Internet about the meaning of words, I'm right and you're wrong. That's just how it is, always. You are probably thinking the same thing, as if you are the keeper of the language, instead of me! (Ok, do we all understand how stupid this flamewar is? Good. Now, let's have the flamewar.)
What you use to define "downloading" many of us call "saving" (or possibly "recording" if the data arrives by means other than IP, such as an OTA TV antenna, camera/microphone, etc). Downloading is what happens in both scenarios. In one scenario, it is "saved" to local media, and in another scenario, it isn't. That second one (where you elect to use the data immediately and don't save it), we call streaming (if the data is AV), but it's viewed as a special case within downloading, not a separate and distinct thing from downloading.
There are a lot of reasons that computer-oriented people use this terminology, but it basically comes down to the fact that we use various programs to get things, where that program doesn't care what happens to the data after it is transported. curl and wget are curl and wget regardless of if you save the data for many years, or just save it to a ramdisk for just a few seconds prior to deleting it, or if you pipe curl right into some other program's stdin eithout ever writing to a file at all. In all these use cases, curl downloaded the data.
That you anti-computer media people would barge into computer peoples' domain and think you can change the meaning of words, has many ha-- oops, you already took "hacker." Well, then: NEVER AGAIN!!