I'm struggling to understand the meaning of $d_1$ and $d_2$ in Black & Scholes formula and why they're different from each other.
As per the formula, $$C = SN(d_1) - e^{-rT}XN(d_2)$$
which means if the call option gets exercised, one would receive the stock and pay the strike price.
Clearly, the strike price payment is conditional on the option finishing in the money, ie. future value of this payment is $X \cdot \mathbb{P} (S_T > X)$ and thus discounted present value would be $$e^{-rT}X \cdot \mathbb{P} (S_T > X)$$
From this we can see that Black & Scholes $N(d_2)$ is the probability of the option exercise (under risk neutral measure).
But then it should logically follow that the first part of the formula - conditional receipt of the stock - should equally depend on the above probability, in which case its future value would be $S e^{rT} \cdot \mathbb{P} (S_T > X)$, with the present value $S \cdot \mathbb{P} (S_T > X)$, and so Black & Scholes formula "should be" $$C = SN(d_2) - e^{-rT}XN(d_2)$$
But the formula does use $N(d_1)$ and since $d_1 > d_2$ my understanding is that it gives higher probability to receiving the stock than to paying the strike price.
I went through Understanding $N(d_1)$ and $N(d_2)$: Risk-Adjusted Probabilities in the Black & Scholes Model and also found this explanation on Quora helpful, but still don't see what is fundamentally wrong with the train of thought I described above.