Detective 26 is cover dated April 1939 and only has "The BATMAN" in text at the top of a humor page, as does Adventure #37 (also cover dated April 1939) and Action 12 is cover dated May 1939, as is Detective 27.
Mike's Amazing World has the newsstand date of Detective #26 as March 21st, Adventure #37 as March 23rd, Action #12 April 4th and Detective #27 as April 18th 1939. So, Detective #26 is the first MENTION of The Batman in writing and Action #12 is the first published PICTURE of The Batman.
The first pic below is from my old copy of Detective #26, the second pic is from my old copy of Adventure #37.