Vasya. Set $x=u+1$, $y=v+1$ and $z=w+1$ to observe that any number not exceeding $N:=10^9-4000$ \[ (u+1)(v+1)(w+1) - (u+1+v+1+w+1) = uvw + uv+uw+vw-2 \]with $u,v,w\ge 100$ is good according to Vasya. Hence, except $N$ and $N-1$, the good numbers of Vasya and Petya are in one-to-one correspondence. Now, I claim $N$ is good according to Vasya. Since $N+2$ is not in the allowed range, this will prove that Vasya has more good numbers. It suffices to observe that $N$ is realized by $(x,y,z)=(10^3-1,10^3+1,10^3)$.

The answer is $\boxed{\text{Vasya}}$. The key observation is that, $$(abc+ab+ac+bc)-2=(a+1)(b+1)(c+1)-(a+1)-(b+1)-(c+1)$$This means that Vasya considers $N$ good iff Petya considers $N+2$ good. It is sufficient to prove that Vasya considers $10^9-4000$ to be good. Noticing $(x,y,z)=(1000,1000,1000)$ is only off by $1000$ trying $(x,y,z)=(999,1000,1001)$ works.

$\boxed{\text{Vasya}}$ has more good numbers. If $x, y, z \geq 101,$ then notice we can rewrite Petya's expression as $(a+1)(b+1)(c+1) - (a+1) - (b+1) - (c+1) + 2 = xyz - x - y - z + 2$ because $a, b, c \geq 100.$ Clearly, because we have an upper bound on good integers in the problem statement, Vasya will have more good numbers.

$a=x+1$ $b=y+1$ and $c=z+1$ expression $abc+ab+bc+ca=-x-y-z+xyz+2>xyz-x-y-z$ hence the answer is $Vasya$