I'm not familiar with your camcorder, but I might make a guess here based on how my camcorder functions.
My guess is the date/time you're seeing in your camcorder's screen is simply a convenience on the display itself. For example, my camcorder will show the date and time, if I so desire, on the LCD display or in the viewfinder, but will NOT stamp the date/time on the video itself unless I set it that way.
But let me ask a stupid question here: Are you sure the date/time you're seeing on the camcorder's screen DURING PLAYBACK is the date/time you shot the video? Or is it the the CURRENT date/time?
Check your camcorder's users manual for date/time display and date/time stamp. I'm not an expert, but I have to believe that if you've indeed stamped the date/time display on the video itself, it will capture along with the shot video.
Here's another thought. Check your camcorder's manual regarding timecode striping on the tape itself. This is digital coding on the tape used "invisibly" by video editing programs during capture and other functions from digital tape.
Personally I find all this date/time stuff confusing while shooting video: what you see in the display versus what you see on the tape can be entirely different things. Again, your camcorder's documentation should help.