Interpreting the two normal terms in Black-Scholes

In the Black-Scholes call price $C = S N(d_1) - K e^{-rT} N(d_2)$, what precisely do $N(d_1)$ and $N(d_2)$ represent?

$N(d_2)$ is the risk-neutral probability the call finishes in the money; $N(d_1)$ is that probability under the stock-as-numeraire measure, and also the delta. So $Ke^{-rT}N(d_2)$ is the present value of paying the strike when you exercise, and $SN(d_1) = e^{-rT}\mathbb{E}^{\mathbb Q}[S_T \mathbf 1_{S_T>K}]$ is the present value of receiving the stock when you exercise. $N(d_1)$ is not the real-world exercise probability.