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?