Can I make discrete recordings on the Apple Watch with Just Press Record?

Yes, Just Press Record will go into the background when you press the Digital Crown and lower your wrist. 

However, watchOS will try to intelligently bring the app it believes you are most likely to want back to the foreground when you raise your wrist.

To prevent this from happening:

1. Press the Crown on your Apple Watch to invoke the app launcher.
2. Launch the Settings app.
3. Tap General.
4. Tap Wake Screen.
5. Toggle off Wake Screen on Wrist Raise.