hot/neutral/ground weirdness with motion sensing switch

Older house (1940s) on the US east coast, trying to replace a simple light switch with a motion sensing switch that wants a ground and got some weird results.

The box has a white, black and bare ground wire coming in. With the breaker off, (A) measuring DC resistance between the white (supposedly neutral) and the ground is near zero as expected, and with super high resistance between any other pair. But with the breaker on, (B) my non-contact voltage tester alerts on the white wire! And the voltage can only be found between the white and ground, not the black and ground. If not for (A) I would think the black and white were reversed. Very confusing.

I tried wiring it both ways but motion sensing switch (Lutron MS-OPS2-WH) only illuminates the LED in the sensor and clicks the relay but the light doesn't come on.