Some colleges say one of any test scores will do ( SAT, IB ) , How true is this ?

One of the universities ( pretty sure its NYU ) mentions that they require one of the following :
a) SAT reasoning
b) SAT subject
c) IB predicted
d) Students may instead elect to submit results from a nationally accredited exam that is considered locally to signify the completion of secondary education and is administered independently of the student’s school.

What I have heard people say is that although colleges mention that they don’t require a certain test score, it is always best to send in the score… How true is this?
What I actually mean is that if a certain applicant has submitted both the SAT and the A level predicted scores , will he have a greater advantage over a student who has only submitted his A level predicted scores.?
In other words, if a particular applicant has submitted scores of many tests ( TOEFL, SAT , SAT subject tests) , will he or she be at a greater advantage than some other applicant who has submitted only ONE of the scores.