It appears like there's a tiny but non-zero difference between the two. P1 and P2 formulas in https://math.stackexchange.com/a/5017119/998
Verification:
Nevermind I found it in your comment on the math.SE post
I can show that expected trace is equal to $(1- \frac{1}{d})^d$ ...
So for large $d$, $(1- \frac{1}{d})^d \to \frac{1}{e}$.
