Reduce $SAT$ to $3SAT$ I was reading this that details a polynomial reduction from $SAT$ to $3SAT$.

In case $k = 1$ or $k = 2$, why don't we just replace those clauses with $C_i' = \{{z_1, y_{i, 1}, y_{i, 2}}\}$ and $C_i' = \{{z_1, z_2, y_{i, 1}}\}$  respectively? Instead, it suggested to replace them by $C_i' = \{(z_1, y_{i, 1}, y_{i, 2}), (z_1, y_{i, 1}, \overline{y_{i, 2}}), (z_1, \overline{y_{i, 1}}, y_{i, 2}), (z_1, \overline{y_{i, 1}}, \overline{y_{i, 2}})\}$ and $C_i' = \{(z_1, z_2, y_{i, 1}), (z_1, z_2, \overline{y_{i, 1}})\}$ respectively.
Let's take the case $k = 1$. Consider the CNF $z_1 \land \lnot z_1$. You suggest replacing it by
$$
(z_1 \lor y_{1,1} \lor y_{1,2}) \land (\lnot z_1 \lor y_{2,1} \lor y_{2,2}).
$$
Hopefully you see the problem now.