Answer :First player will win. Proof: Claim: A player who made piles like N and N-1 , has a winning strategy. "Proof" of the claim: Just do strong induction. Back to the problem . The first player will transfer 674×2 stones to the other pile . And the piles will be this form: (673,674), by the Induction first player wins.

Here's a cleaner write-up , denote the players by $A, B$ according to their names and let $(\bullet, \bullet')$ denote the sizes of boxes so that it's $(2021, 0)$ initially. Claim. B has a winning strategy. (for now, assume it's false) Subclaim. The only final position someone can possibly lose at is $(0, 1)$ Proof for the subclaim . Notice the invariance of the difference of the sizes of boxes modulo $3$. $\square$ Proof for the claim. Now, we give a strategy for B, in particular we claim that he can always leave a $(k-1, k)$ position to A starting with $$(2021, 0)\to (673, 674)$$and by playing symmetrically wrt him afterwards. It's easy to check that it can always be done as long as A can make a move . $\square$ So, since $(2, 0)\to_{A} (0, 1)$ is the only possible move A can make to win, according to Bogdan's strategy he can prevent it and win .

Bogdan will win. Let $(m,n)$ denote the number of stones in the two piles. Define $W=\{(a,b)|a,b>0,|a-b|\ge 2\}, L=\{(a,b)|a,b>0,|a-b|< 2\}$. Claim. A player has a winning strategy if and only if he gets a position in set $W$. Proof. To prove the claim, (1) and (2) is needed. (1) For any element in set $W$, a player can operate so that after the operation, the position is in set $L$. (2) For any element in set $L$, after any possible operation, the position is in set $W$. (1). Assume $(a,b)\in W$,and $a-b\ge 2$. The operation $(a,b) \to (a-2[\frac{a-b+1}{3}],b+[\frac{a-b+1}{3}])$ satisfies. (2). Enumerate all the possible stiuations. So using (1) and (2), the position each player get can only be in one of the two sets, and because $(1,0),(0,1),(1,1)$ will cause losing, so the one gets a position in the set $L$ loses, and the one gets a position in the set $W$ wins. Notice $(2021,0)\in W$. Thus the first player, who is Bogdan, will win the game.