Understanding PDA for odd length string with middle symbol 0

I came across this pdf, which describes the language of odd length string with middle symbol 0 as follows:

enter image description here

Doubts:

  1. I dont understand the transition labels. In standard resources like books by Ullman et al, Linz and in wikipedia, the transition labels take following form:

    • $ a,b/ab$ means if next input symbol is $ a$ and current stack top is $ b$ , then push $ a$ on $ b$
    • $ a,b/\epsilon$ means if next input symbol is $ a$ and current stack top is $ b$ , then pop $ b$
    • $ a,b/a$ means if next input symbol is $ a$ and current stack top is $ b$ , then pop $ b$ and push $ a$

    I dont get meaning of transition labels in diagram $ a,b\rightarrow c$ . Some one explained me that its, if next next input symbol is $ a$ , pop $ b$ and push $ c$ . I feel, if this interpretation is correct, then this notation is insufficient as it will describe both $ a,b/ab$ and $ a,c/ac$ as $ a,\epsilon\rightarrow a$ . Am I right with this, or I understood the notation incorrectly?

  2. Assuming above interpretation to be correct, loop on $ q_1$ pushes all input symbols, be it 1 or 0. Then for $ 0$ at any position (not necessarily middle position), it transits to $ q_2$ . Loop at $ q_2$ pops all symbols. I dont get how above PDA forces middle symbol to be $ 0$ . Also I dont get how it ensures length of $ w$ is odd.

  3. If given PDA is incorrect, can we prepare correct one by re-labelling as follows:

    • Loop at $ q_0$ : $ \{(1,$ /1);(0,$ /1);(0,0/00);(0,1/01);(1,0/10);(0,1/01)\}$
    • Transition $ q_0-q_1$ : $ \{(0,0/0);(0,1/1)\}$
    • Loop at $ q_2$ : $ \{(0,0/\epsilon);(0,1/\epsilon);(1,0/\epsilon);(1,1/\epsilon)\}$

    So, its CFL not deterministic CFL, right?