Hi Hayley,
Just putting my spoke in the wheel.
I tested date stamp in Numbers by formatting Column B as:
Format Panel > Cell > Data Format > Date & Time > Date (in my Region, weekday, day of month, year) and Time as hour, minute, second, am/pm.
I did that to test how Numbers handles a date stamp. I included Time so that I won't have to wait until tomorrow to find out if it is an automatic update like the NOW or TODAY functions.
Cell B2 Menu > Table > Insert Current Date works as intended. The date is correct, and the Time is 12:00:00 am (midnight at the start of that day).
Cell B3 Menu > Table > Insert Current Time works as intended.
Cell B4 (some minutes and seconds later) Menu > Table > Insert Current Time works as intended.
My test confirmed that Insert Current Time is a Date & Time stamp that does not affect previous Date & Time stamps in the way that the functions NOW and TODAY automatically update.

If the Date part works for you, you don't need to display the Time part.
Regards,
Ian.