Okay, I took the advice and re-read my owner's manual and found that
Suzuki calls the two tiny holes I described earlier the "pilot
holes." And yes, they are what others have been referring to as the
pee hole(s) and/or tell-tale hole(s) as they serve the same purpose.

According to the manual, when water is squirting out of these two
little holes while the motor is running, it indicates the cooling
system is working properly. So it sounds like that mystery is
solved. Thanks again for the reminder.

-Jay