Why does the fluid inside a cup not spin when the cup is spinning, but starts to spin when the cup stops spinning?

I experimented with a toothpick in a cup of tea. You are wrong about the way the liquid spins after the cup stops rotating. It follows the rotation.

The fluid in the cup acts as a body separate from the cup. The turns you give to the cup because of the weak adhesiveness of water to the sides, is not transmitted to the fluid.

By turning the cup angular momentum is generated . When the cup stops moving conservation of angular momentum requires that it is transmitted somewhere. In this case part of it will be transmitted and neutralized by the table top but part of it goes to the liquid by the weak adherence it has on the sides.