Here's a screenshot of credits after a TV broadcast of a Japanese version of a Broadway musical called "A Night with Janis Joplin".
The whole credits were divided into two distinct sections: one in Japanese related to the TV broadcast by WOWOW, the second in English, clearly relating to the live staging by the Broadway Asia, so I strongly believe the problem in my question pertains to the U.S./international law rather than being Japanese-specific (or a simple translation issue).
The English subsection MUSIC CREDITS lists all the songs used in the musical with their due credits.
After each song, there is a phrase "used by permission" or "used with permission" and sometimes the two are combined (!).
Here's a sample screenshot for four songs (one staged as a medley):
What is the legal difference between the phrases "used by permission" and "used with permission"?