EVERYTHING you look at on your computer or smartphone or tablet is download first, no matter if you are downloading or streaming.
But you say "wi-fi data allowance" and I am not sure you understand the difference between "wi-fi" and "data allowance" (on say a smartphone or tablet)
WI-FI; If you are at HOME and using wi-fi on your own broadband then anything you download costs you no extra (unless you have a limit on your broadband usage). You can download as much as you like and pay no extra.
If you are in a pub or restaurant or hotel and using THEIR Wi-Fi this also costs you nothing to download or stream TV programs or films.
However if you are out and about and NOT connected to Wi-Fi then anything you download on your smartphone or tablet will eat into your "data allowance".
Every contract for a smartphone or tablet will have a limit on your data download. If you go over you may have to pay extra.
But if you are at home, using you own broadband, and your own Wi-Fi, and you pay for "unlimited broadband" then you can watch as many TV programs or films as you like and it will cost you no extra.