[orless]

a x b = b + ... +b is not a strict definition, it's just a convention.

Check the English Wikipedia:

https://en.wikipedia.org/wiki/Multiplication

It's, as you say, 5x3 = 3 + 3 + 3 + 3 +3.

But now check Russian Wikipedia:

https://ru.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BD%D0%BE%D0%B6...

There you'll see 5x3 = 5 + 5 + 5.

So much for the "very well-defined mathematical meaning".