When you keep pointing back to "a x b = b+b...+b", as The Definition without including the properties, it means you're mixing up the orthography[0] of multiplication with the real underlying idea of multiplication.

A math definition includes that all properties must simultaneously be true. It's the limitations of writing (orthography[0]) that we state things one thing before the other. The phrase "and then" used as a sequential condition is not applicable. Instead, if all properties are true, you thus have the definition.

Here's another "definition"[1] that states the summation in reverse order: "In simple algebra, multiplication is the process of calculating the result when a number a is taken b times."

e.g. "when a number 5 is taken 3 times" ... which is the repeated addition the child carried out.

That wikipedia stated multiplication as "a x b = b+b...+b" while Wolfram MathWorld stated it as "a is taken b times." is a difference in orthography and not definition. Unfortunately, you're working backward from an arbitrary orthography and judging the child to be wrong.

[0]https://en.wikipedia.org/wiki/Orthography

[1]https://books.google.com/books?id=aFDWuZZslUUC&pg=PA1974&lpg...

The contents of the Weisstein book was also used in Wolfram MathWorld:

[2]http://mathworld.wolfram.com/Multiplication.html