Watching streaming video is downloading, but it's not the type of downloading that leaves you with a file that you can play again offline.
To see anything that is on the internet, you need to download it.
When you type a website's name into the url box at the top of the browser, this gets sent to a dns server that looks up the ip address of the said site. For example, if you type "www.horror.com", this is looked up by a dns server and the dns server says "OK,
www.horror.com is at xxx.xxx.xxx.xxx" and directs you as such. (sort of like looking up someone's number or address in the phone book when you know their name).
So, basically then a request is sent for the page that you surfed to be downloaded to your computer. In the process, you have uploaded some data to the internet (ie you have sent a request from your computer to the website that you want to view, which is a small upload of data).
The information that you request is downloaded so that you can view it. It may not be stored in a way that has much permanency (eg it may go to a temp folder that is deleted after the session or at some other time).
Similarly, if you wish to watch a youtube video, you have to download the data that the video is comprised of. Because it is a streaming video, you will view it whilst it is still downloading. If you connection isn't fast enough, it may pause or stutter, whilst it is "buffering" more data. Again, this downloaded data won't be stored in a way that you can watch it again offline later. But it is certainly downloaded.
If you live in a country like Australia where we have quotas on monthly downloads, your ISP will probably offer an online usage monitor (so you can see how much you have left for the month). Watching youtube or other streaming video certainly uses up your quota, as the data that comprises the stream has to be downloaded to be viewed. (Surfing to websites and reading email also uses quota as these are also downloads, but usually small compared to streaming video).
So, basically everything you view on the net requires a download (and a - usually - much smaller upload to request the info in the first place).
One obvious difference between a streaming video and a video file that can be played offline is the temporary nature of the storage of the first set of data vs the persistent nature of the storage of the latter. But they are both equally a "download"