But didn't we execute (or at least prosecute) the Japs for the same thing?

After World War II, we convicted several Japanese soldiers for waterboarding American and Allied prisoners of war. (Washington Post)

In 1947, the U.S. charged a Japanese officer, Yukio Asano, with war crimes for waterboarding a U.S. civilian. Asano was sentenced to 15 years of hard labor. (NPR)

Fifteen years of hard labor for waterboarding? Not exactly. These reports purposely omit the other guilty charges found against Yukio Asano : "by beating and kicking them, by forcing water into their mouths and noses; and by pressing lighted cigarettes against their bodies." Or listed here: beating using hands, fists, club; kicking; water torture; burning using cigarettes; strapping on a stretcher head downward.

And what about the claim Japanese soldiers were sentenced to death by American judges for waterboarding? Is there any evidence of Japanese being executed specifically for waterboarding?

Here, at any rate, is Japanese style waterboarding:

There are two forms of water torture.
In the first, the victim was tied or held down on his back and a cloth placed over his nose and mouth. Waters was then poured on the cloth. Interrogation proceeded and the victim was beaten if he did not reply. As he opened his mouth to breathe or answer questions, water went down his throat until he could hold no more. Sometimes he was then beaten over his distended stomach, sometimes a Japanese jumped or sometimes pressed it with his foot.
In the second,
The victim was tied lengthways on a ladder, face upwards with a rung of the ladder across his throat and his head below the ladder. In this position he was slid head first into to a tub of water and kept there until almost drowned. After being revived, interrogation continued and he would be re-immersed.

Quite a difference from our version of waterboarding, wouldn't you say? In fact, the three we waterboarded were assured before hand that it would not kill them.