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Come on, don't tell me Spotify can't code that. It sounds like the most simple thing in the world.
It seems there used to be an option for that which is now gone, why? Probably because of spotify's developers ineptitude.
Thanks for getting in touch.
When you click a Spotify link, it should ask you if you'd like to listen on Spotify. After clicking Yes it should open Spotify.
Perhaps try reinstalling Spotify. You can read here how. Reboot the computer after doing this.
Try both using the link when Spotify is not open, and when Spotify is open.
Hope it works. Let us know how it goes. Have a nice day!