It may seem like a small difference, but it's pretty large in practice.
> willfully and contrary to such oath states or subscribes any material matter which he does not believe to be true
Intent (“willfully”) matters.
If you believe the statement is true when you say it, it isn’t perjury. It’s just incorrect.
Yes, but whether you believed the statement was true when you said is not just a matter of your saying so. Other evidence can be brought to bear on the question: for example, as I have said elsewhere in this discussion, asking you how you came to believe that what you said was true, and you either admitting that you just made it up or had no good reason to believe it, or giving obviously false or evasive answers, or answers that are rebutted by other evidence (for example, you say you were told it by someone who denies that they did so). That could lead a jury in a perjury trial to conclude that in fact you were wilfully lying, even though you deny it.
But it's still testimony given under oath. AFAIK perjury is still applicable under those circumstances.