Time Stamp data is stored internally as an integer that counts the number of seconds since an early arbitrary date and time. Thus, the actual data stored is the same regardless of system settings. The system settings affect data entry and data display (if you don't specify a format for the field on your layout.)
That's why Set field appears to work here. It computes the integer and this will be the same on both system. The insert option, fails probably due to the first bug listed below.
I can find several different bug reports regarding issues with system date settings and Filemaker in the Known Bug List:
Only the first link (and possibly the second) seems specifically relevant to your issue.
Thanks for the answer.
It still doesn't explain why two identical databases on the same server behaves differently. The only thing I can think of is that the machine that was used for cloning the first database was set to the wrong date format (which is doubtful).