Primary display setting has no effect

I'm using 15.3 Lite on a Dell laptop that's connected to a Samsung monitor via HDMI. I want to switch to using the monitor as my primary display, so the task bar shows up on it. However, even after going into Settings -> Display -> General and setting "Primary Display" for the monitor to on and clicking "Apply", nothing happens. The setting is saved and it stays that way even after restarting, but everything still shows up on the laptop screen and the monitor is just the second display.

Is this something that can be checked or am I misunderstanding the use of the "Primary Display" feature?

The dual monitor setup page on the Ubuntu website states:

The primary display is the one with the top bar, and where the Activities overview is shown.

I would expect that to refer to the task bar over in Zorin OS.

This is a very good question. I ran into this same trouble on a Notebook and External Monitor and my solution, in the end, was to switch the positions of each on my desk.
Setting up multiple monitors is easy and works quite well with the default GUI Display settings. However, using a Notebook as a monitor adds a certain complication. A notebook is intended to be portable and relies on its Built In Screen display, so the hardware is built to only default to Its Built In Display. Getting around this is not easy and 'how to do it' varies a lot based on the manufacturer.
It's easy to understand why a notebook manufacturer would want to do this. The built in display is to spec and should operate a certain way and last a certain amount of time. An external Monitor can be a wild card and if the external display is the Primary display and it goes out, the user can lose access to their notebook and be unable to easily do anything about it. The manufacturer does not want to invest in Call Centers sorting out problems with the device the manufacturer had no hand in the making of.