Do c sections happen when a baby is late?

 Usually but the real underlying factor is other issues zx

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No that's not how it works

C-sections happen for a lot of reasons.  They can be done:

-Early, because something isn't going right with the baby or with mom and they need to get the kid out quickly to fix that

-Right about on time or just a whiff early.  While c-sections without a medical indication are rare, they do happen.  And a woman who has had previous c-section deliveries will often opt to just schedule the next one for term rather than try for vaginal birth.

-Late, if the mother doesn't go into labor on her own and other methods of getting the kid out aren't working

Your mom almost certainly knows if you were early or late.

A c-section happens when there is a medical reason that vaginal birth will be dangerous for the baby or the mother.

If a mother has had health issues during the pregnancy, such as high blood pressure or gestational diabetes, a c-section is likely.  If the baby is in the breach position, a c-section will be likely.  

A c-section can also happen if complications happen during labor.  If normal labor is not progressing or if the baby goes into distress during labor or if the mother has a medical complication during labor, then a c-section would be done.

Just being "late" is not a reason. My second baby was two weeks late - but I did not have a c-section.

No. C sections are performed when there might be potential harm to the baby or you. If your birth canal is too small, then a c might be performed. 

It has nothing to due with being late or early. 

typically yes my baby was due on December 30, and I had a c section on January 3

Possibly, but not necessarily.  There are other reasons.