A verdict of "not guilty" is rendered in American courts when the judge or jury (some defendants waive their right to a jury trial and are tried by the judge) concludes that the prosecutor has not sustained his burden of proof. That burden of proof requires a showing of every element of the charged offense beyond a reasonable doubt. The presumption of innocence is a legal presumption which entitles the defendant to require the prosecutor to meet the stringent standard of proof. It is not a factual determination of innocence, but a standard of legal innocence.

Occasionally, a judge or jury really does conclude that the defendant didn't do it; but I venture to say that most (90+%?) of the time, a not guilty verdict is the result of the failure to establish guilt by the requisite amount of evidence. Defendants who claim they have shown their innocence are practicing spin control.

In a civil case, the plaintiff's burden of proof is typically that of "preponderance of the evidence," a much lesser standard. That's why you can have a criminal case with a not guilty verdict but a civil case out of the same circumstance which imposes liability. OJ Simpson was not innocent, he was not guilty; and in the civil trial, he was liable.

Various burdens of proof range from preponderance of the evidence (ie, more evidence for the proposition than against it -> 51% ish?) to clear and convincing evidence (80%ish?) to beyond a reasonable doubt (99% ish?).