According to my battle.net account I own StarCraft II: Legacy of the Void. There's a download button for me to get my copy if I want.
However, I'd prefer to download and play it through the Blizzard desktop app. When I install that and go to the StarCraft II section it says it's the "starter edition" and there's a "Try for Free" button.
I cannot see any way to use the client to play Legacy of the Void. There's nothing in the drop-down menu for regions (which I understand is where it was during the beta). There doesn't seem to be any more recent advice for this that I can find.
Is it possible? If so, how?