I'm using BOINC Manager v7.8.3 64-bit on Windows 10 Pro 64-bit Spring Update. When I exit (X out) of BOINC on my desktop, I expected all processing to stop. However, after initializing BOINC again, I find that several projects have gone to completion, meaning they weren't completed when exiting but now are.

Is BOINC running these projects in a background service or something?

Use File->Exit on BOINC Manager, or right-click the BOINC icon and use Exit, then use "Stop running tasks when exiting the BOINC Manager" to stop processing.
If you do not have the exit dialog, you can enable it in BOINC Manager->Options->Other options->General->Enable Manager exit dialog?

Pressing the X in the top right corner will only minimize BOINC Manager to the system tray, it will not exit it.

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If by 'X out', you mean the red cross in the top-right corner, this is by design. The cross removes the Manager window from your screen, so can use the space for something else, but doesn't fully close it: instead, it's minimised to the 'notification area' or 'system tray' at the bottom right (by the clock).

BOINC itself does very little processing, and can safely be left running while you work. It's the scientific research applications that BOINC manages which can sometimes get in the way and slow things down - although the effect is designed to be minimal.

If you need to stop the science applications from running, there are various options you can use. The default settings are to suspend computation when the computer is 'in use': that means when activity is detected with the mouse or keyboard. So, it doesn't work if you're just passively watching a film or something.

It is possible to set BOINC to stop processing when it detects that a specified application is running on the computer: that can be useful if you play computer games or have an intensive business application.

And finally, you can exit BOINC completely (closing the science applications as you do so) by selecting 'Exit BOINC Manager' from the File menu at the top left of BOINC Manager. This is probably closest to your original idea, but have a look round the other options and ask again if you have a specific need.