Is it Itsy Bitsy Spider or Incy Wincy Spider?
Both are right. It is the same rhyme with a different name: Americans usually sing “Itsy Bitsy” and families in Britain and Australia usually sing “Incy Wincy”. You will also hear “Eensy Weensy”.
Do the words change too?
Barely. The story stays the same: spider climbs, rain washes the spider out, sun dries the rain, spider climbs again. Small words shift from family to family, so don’t worry if your version isn’t quite the same as the one a friend sings.
Which one should you teach?
Use the name your child will hear at nursery or school. If your family grew up with the other one, teach both. Two names for one spider tends to strike small children as very funny, and noticing the difference is good listening practice.