@drearyweather: That is indeed correct. They own the rights to "the musical composition", not the song or its performance.
Sound recordings and musical compositions are considered two separate works for copyright purposes. Even though a sound recording is a derivative work of the underlying musical composition, a copyright in a sound recording is not the same as, or a substitute for, copyright in the underlying musical composition.
They have worded their argument very specifically. They want the singer to stop the distribution of the sound recording and public performance, by claiming they have the rights to the musical composition. This is so antithetical to copyright as it would mean that people that create sheet music of eg. Beethoven's fifth would suddenly own all the performances as well. It would also mean that the congregations could be liable for the few cents (51 cents I believe) per person it costs to license whenever they sing the song, actually, legally speaking, if they did not collect said license from their congregations, they may lose their rights to the work altogether.
You can buy the musical composition of many songs without buying the copyright on the further distribution and performance. You CAN buy public performance rights as well, but that is typically a ton more expensive and separate from the piece of sheet music from a composer. This is what makes the specific wording so suspicious, it points to a fact they do not own those rights, which means they did not write it themselves but purchased it from someone else. If you create a piece of music yourself, you can claim the performance rights, which they do not in the suit.
To give you an example, many (most) singers do not write their own songs, they buy them from someone, the bigger musicians will also buy the exclusive rights and the big labels will also pay out to get the exclusive rights on the public performance. Hence you cannot perform Beyonce's latest hit album and sell it, hell, they'll kick you off YouTube and fine you for just humming a snippet of it online. Same goes for Happy Birthday, licensed to I believe BMG or Warner Music, you can't perform it in public without paying them.