Hi WinoGunner,
Thank you for reaching out Microsoft Community.
You can use either of the two. A mini-DP to HDMI adapter will do or a mini-DP cable to directly extend your Surface display to an external monitor as long as the DP at the back of this monitor is active. You can check this link https://support.microsoft.com/en-us/help/4023496/surface-troubleshoot-connecting-to-a-second-screen for reference. You'll find out external monitor models that were tested to work just fine using an adapter or a Surface dock.
If this answers your question, feel free to mark this post as an answer. Otherwise, don't hesitate to respond to this thread if you need further assistance.
Kind Regards,
Crest_Z