Chord names can be a really confusing aspect of music theory because there is almost always more than one correct name for a chord and which name is used in a particular situation depends somewhat on personal preference. For example, a “minor MAJOR 7th” could also be described as a “MAJOR 7th/b3” and a “MAJOR minor 7th” could also be described as a “minor 7th/#7”.
It’s about as clear as mud.
But, when in doubt, simply work through the parts of the name that are most descriptive and think in terms of precedence. The key is to look for the type (MAJOR, minor, Augmented, diminished) and then the number suffix.
minor (MAJOR 7th) would mean take a MAJOR 7th and leave the 7th in tact, but flatten the 3rd so that it resembles the 3rd in a minor triad. Or, it could mean to take a minor 7th and sharpen the 7th so that it resembles a MAJOR 7th.
To me, the first example is the most descriptive. It basically says to take a MAJOR 7th and retain all of it’s 7’thy goodness but make the triad part of the chord a minor. In other words, flatten the 2nd note of the MAJOR 7th chord but leave the rest of it alone.
Any guesses as to what a MAJOR 7th/b7 would be?
The answer to that and more can be found in the section on 7ths.