As always, I am in awe of the collective knowledge shown by the posters here.  I withdraw my comment that Schiff copied Weisinger--clearly the facts are otherwise.  However, as a kid and as an adult, I have always had the reaction that when Batman and Superman had the same kind of stories--transformations, imaginary stories, whatever--the Superman versions were better, and the gimmicks that worked with Superman just were not as effective with Batman.

For that reason, I was ready for the "New Look" to come along by mid-1964.  I know a lot of you do not agree, since moving Infantino onto Batman came at the price of ending the classic era of Adam Strange stories.  But still, even down to the cover art, things got better for Batman once the New Look came in:

 Batman_160.jpg   Batman-166_zps130d500d.jpg

One other note:  I went back to Mike's Amazing World to check on when in 1956 Batwoman came along.  I was surprised to find that she arrived two months before the Flash appeared in Showcase #4.  That means that even though the Silver Age is generally counted as having started with Schwartz's revival of the Flash, Schiff had originated two DC Silver Age heroes --Batwoman and Jonn Jonzz--before him.