I would like to add the following:

I think the source of confusion is the fact that the Killing form is nondegenerate (for semi-simple Lie groups) and negative definite (stronger than non-degenerate) for compact Lie groups with trivial center

$SL(2)$ is semi-simple but not compact.

The torus $S^1$ is compact but not semi-simple (abelian).

Compact groups are reductive and semi-simple only when in the case of trivial center.

I would like to add the following:

I think the source of confusion is the fact that the Killing form is nondegenerate (for semi-simple Lie groups) and negative definite (stronger than non-degenerate) for compact Lie groups with discrete center

$SL(2)$ is semi-simple but not compact.

The torus $S^1$ is compact but not semi-simple (abelian).

Compact groups are reductive and semi-simple only when in the case of discrete center.