Stress tensor of fluid in equilibrium, inertial frame

There are some points in this wikipedia chapter. Main equation is:

$$ T^{alpha beta} , = left(rho + {p over c^2}right)u^{alpha}u^{beta} + p g^{alpha beta} $$

where $c$ is explicit.

The one for the trace is:

$$T = 3p – rho c^2$$

that seems contradictory with:

$$T^{alphabeta} = left( begin{matrix}
rho & 0 & 0 & 0
0 & p & 0 & 0
0 & 0 & p & 0
0 & 0 & 0 & p
end{matrix} right)$$

with trace $3p+rho$ (difference in sign and value of last term).

The expression for the four-velocity:

$$u^{alpha} = (1, 0, 0, 0)$$

is not the usual one $(c, 0, 0, 0)$.

Finally, the metric:

$$g^{alphabeta} , = left( begin{matrix}
– c^{-2} & 0 & 0 & 0
0 & 1 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1
end{matrix} right)$$

also with explicit $c$, it is also not the usual:

$$left( begin{matrix}
-1 & 0 & 0 & 0
0 & 1 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & 0 & 1
end{matrix} right)$$

Are the wikipedia equations in this chapter using a coherent notation ? If yes, how to explain the previous points ?