Question about Other names -- one letter difference

[s3nr1]

So here's the story,

Our family came to Canada when I was about 16 and we went back to our home country when I was about 23. Around that period I completed Highschool and College.

Here's the problem, back home we don't really have a last name, it's more like first and middle name, so in our passports, we got our Dad's middle name as our last names. And what's more, the last name was misspelled. Which just stuck, it's in my highschool and college diploma.

E.g.
(original as in birth cert)
Dad: Dad Dadname
Me: Greg Gregname

(our passports)
Dad: Dad Dajmame
Me: Greg Gregname Dajname

(new passport when coming back to Canada to continue school)
Me: Greg Gregname Dadname (to correct the mistake, as it were)

I put this name discrepancy as another name in category "Other" and the application asking for a document.

Document: Legal Name Change Doc.
You indicated you have another name. You must provide information on the use of this other name (such as a letter of explanation, legal documents of a name change, etc).​

The question, what kind of document is best here? It's not exactly a name change. Should I even declare this as another name?

Thanks!