Are Schwarzschild radius and event horizon the same thing?

[PeterDonis]

as spacelike 2-surface it actually represents '2-sphere space' at Kruskal coordinate time $T_0$ and $X=T_0$ (i.e. the subset of events simultaneous in KS coordinate chart having $X=T=T_0$), I believe.

Yes. Every point on the Kruskal chart represents a spacelike 2-sphere.