Why do the summoned Servants all belong to different classes?
I think you answered this yourself:


  Using relics and additional incantations during the summoning ritual,
  Masters have some degree of control over the Class


The keyword here is "some". The Servants are summoned by the Grail, even though the Masters have some sort of control of what they are trying to summon. This wiki states (it has a reference too, but the reference is in Japanese):


  The Three Knight classes are the only Servants who will definitely be
  summoned according to the framework of the Holy Grail War, allowing
  for variation in the rest of the classes. More than one version of the
  same class cannot be summoned, even if all the Masters were to use the
  variation in the summoning chant to get a Berserker-class Servant.


So apparently in this case, the Grail will summon a Servant based on it's compatibility with the concrete Master, much as it has happened with


   Caster's master Ryuunosuke Uryuu