Different boolean degrees polynomially related? Let $f$ be a Boolean function.

Let $p$ be the minimum degree real polynomial that represents $f$ with degree $d_f$. 

Let $p_\epsilon$ be the minimum degree real polynomial with degree $d_{f,\epsilon}$ such that $$|p_\epsilon(x)-f(x)|\leq\epsilon.$$

Let $p_{0,\epsilon}$ be the minimum degree real polynomial with degree $d_{0,f,\epsilon}$ such that $$f(x)=0\implies p_{0,\epsilon}(x)=0$$$$f(x)=1\implies|p_{0,\epsilon}(x)-f(x)|\leq\epsilon.$$

Let $p_{1,\epsilon}$ be the minimum degree real polynomial with degree $d_{1,f,\epsilon}$ such that $$f(x)=1\implies p_{1,\epsilon}(x)=1$$$$f(x)=0\implies|p_{1,\epsilon}(x)-f(x)|\leq\epsilon.$$

Are $d_f,d_{f,\epsilon},d_{0,f,\epsilon}$ and $d_{1,f,\epsilon}$ all polynomially related?
It is clear $$d_{f,\epsilon}\leq d_{0,f,\epsilon},d_{1,f,\epsilon}\leq d_f.$$

Does $$d_{f}\leq (d_{0,f,\epsilon})^a,(d_{1,f,\epsilon})^b\leq (d_{f,\epsilon})^c$$ hold for some $a,b,c\in\Bbb N$?

What is a good reference to understand relations among these four degrees?
It is known that
$$ d_f = O_\epsilon(d_{f,\epsilon}^6), $$
for all $\epsilon &lt; 1/2$. This is described in many lecture notes, for example these quantum lecture notes by Montanaro (Theorem 3). It is suspected that the best exponent is $2$. It is known that the exact value of $\epsilon$ only affects the constant.

Since $d_{f,\epsilon} \leq d_{0,f,\epsilon},d_{1,f,\epsilon} \leq d_f$, the other measures are also polynomially related.