On the machine that doesn't have any problems ...

Let me see if I've got this right.

On both machines you can run:

perl -MDBI=":sql_types" -le "print 1;"
[download]

and, on both machines, the following outputs 1.637:

perl -MDBI=":sql_types" -le "print $DBI::VERSION;"
[download]

But you can run the following command only on the machine that doesn't have any problems:

perl -MDBI=":sql_types" -le "print SQL_GUID;"
[download]

I think Strawberry-5.20.3 shipped with version 1.634 of DBI (at least that's the version I have), and I think you'll still have that version installed in vendor/lib.
I would try reverting to that version - which you could do by renaming site/lib/DBI.pm to site/lib/DBI.pm_hide and site/lib/auto/DBI/DBI.xs.dll to site/lib/auto/DBI/DBI.xs.dll_hide.
If that fixes the problem (with the last of those one-liners) on the problem machine then it might give us something to work on.

Just rename those 2 files to their original names when you wish to revert back to using 1.637.

Cheers,
Rob