Mathematicians have the same definition of factor as what you used. It is your definition of repeated factor that I was confused about. To me if f divides n, and f divides n/f, then f is a repeated factor of n. It seems from your description that to you say that f is a repeated factor of n if and only if f*f=n.

That isn't confusing in a "non-mathematician vs mathematician" way. That is confusing in a "you used a phrase that described what you meant without noticing that it was very ambiguous" way. That is, I am confident that I would not have understood that as you intended back in highschool. And I was likewise not confused when I first saw the phrase "this factor is repeated 3 times".

In ths case your reasoning has a non-obvious step from "odd number of factors" to "one factor must be the square root of the number". It is easy enough to fill that in - just add a comment that you can pair off factors by putting i with j if i*j=n, and you will have a left-over if and only if there is a factor i with i*i=n. But since that pairing is the key step, it needs to actually be stated. You can't expect other people to have that flash of brilliance.