Phone takes videos instead of pictures in night mode

Did you try restarting the phone as I suggested? With Live Photo turned off, as you do, and the camera set to Photo, unless there is something wrong with your iOS, then it cannot shoot a 2 hundreths of a second video. Besides that, you do realize how quick .02 seconds is. I suggest that you use this to make a Genius Bar appointment at an Apple Store near you to have someone from Apple take a look at that device. Genius Bar Reservation and Apple Support Options - Apple

The iPhone cannot take a video in Night Mode. When you see the timer, that is how long the shutter stays open when taking the photo to get extra light on the sensor, and you have to keep the device still. If you check what you took, it is not a video. If you are seeing a smear, then that is because you moved. If you have Live Photo turned on, you will take a short video, but the device cannot be in Live Photo and Night Mode at the same time.

If those are the settings that you were using, there is no way for it to take a video. Try powering your iPhone off and back on again. I just tested my iPhone will a similar set of settings, and all I got was a photo.

Without showing any personal information, please post a screenshot of exactly what icons you are seeing on the screen when attempting to take this picture and receiving a video.