You are wrong.
If Morrissey was to write lyrics and Boz wrote the music then both would be credited as SONGWRITERS for that track, right?
To my knowledge, Morrissey writes the melodies he sings. A pure lyricist in the Broadway or Tin Pan Alley sense would write lyrics to go along with an already-composed melody.
Whether that makes Morrissey a song “songwriter” is debatable but he’s more that just a lyricist.