My name on transcript has no surname. Should I list it as a "former name" when applying for universities in U.S.A. and Canada?
It sounds like the forms you have been provided already give a sufficient means of clarifying, particularly you indicate the form is specifically meant to link a current full name to an alternative shown on formal records.  While the university may have been thinking about different cases, like names changing via marriage, it fits your situation as well.

I would suspect getting an affidavit or other formal attestation is not necessary: if they are not asking for it in the other cases where a name does not match, they should not expect it for your case either.