Overbar as counter in a loop

I want to create a table with four different counters that I want to call $a, bar{a}, b, bar{b}$. For that my obvious code is

$$text{Table}left[pleft[a,bar{a},b,bar{b}right],{a,0,1},left{bar{a},0,1right},{b,0,1},left{bar{b},0,1right}right] // text{Grid}$$

which returns

{{p[0,0,0,0],p[0,1,0,1]},{p[0,1,1,0],p[0,1,1,1]}}   {{p[0,0,0,0],p[0,1,0,1]},{p[0,1,1,0],p[0,1,1,1]}}
{{p[1,0,0,0],p[1,0,0,1]},{p[1,0,1,0],p[1,1,1,1]}}   {{p[1,1,0,0],p[1,1,0,1]},{p[1,0,1,0],p[1,1,1,1]}}

However a quick look shows that this is not the expected result – elements in the first row are repeated. If I write

Table[p[a, c, b, d] , {a, 0, 1}, {c, 0, 1}, {b, 0, 1}, {d, 0, 1}] // Grid

instead then I get the desired output. So my question is double:

1. Why does it happen?
2. How can it be fixed? That is, can I use variables such as $bar{a}$ or $a’$ or $a_1$ for loops modifying something or I am forced to used other names which do not involve these symbols?

SetAttributes[OverBar,HoldFirst]

t1 = Table[
    p[a, OverBar[a], b, OverBar[b]],
    {a, 0, 1}, {OverBar[a], 0, 1}, {b, 0, 1}, {OverBar[b], 0, 1}
];
t2 = Table[p[a, c, b, d], {a, 0, 1}, {c, 0, 1}, {b, 0, 1}, {d, 0, 1}];

t1 === t2