There are a number of different fields used to store the artist:
Artist : Contains the artist as credited on the album cover
Album Artist: If song on a multiple artist release is the artist that the release is credited to.
Sort Artist:A version of the artist that makes it easier sort by moving commons words such as ‘the’
Artists:Used by Jaikoz to allow one song to be credited to multiple artists.
Under Windows/Mac it will look at the Artist field, but I think it might use the Sort Artist or Album Artist field if the Artist field is missing. When you fix songs from Musicbrainz it should use the same value each time, probably what is happening is that songs are being fixed and some songs are not which is why you are getting the difference. You can use Edit/Set Value within Jaikoz to easily set all the selected fields to the same value.
The band official name is called The Beatlesand its sort name is Beatles, The. If I were you I wouldn’t try and change it name to Beatles because this is incorrect and will cause you additional work whenever you fix more Beatles songs. Instead take advantage of the Sort Artist field which allow you to the list ‘The Beatles’ under B rather than T