Issue understanding the reduction of SAT to 3-SAT in poly time
The clauses are not intended to be equivalent, so you shouldn't call it an "equivalent representation in 3-literal clauses."

All the reduction requires is that the original formula has at least one satisfying assignment if, and only if, the new formula does. In this case, we have this property: if the original formula was satisfiable, we can choose at least one assignment to the variables $y_i$ that makes the new formula true and it doesn't matter that there are also choices for the $y_i$ that make it false. Conversely, if the original formula was unsatisfiable, no choice of the $y_i$ will make the new formula true.

Formally, there doesn't even have to be any relationship between the values of the variables in the two assignments. In theory, you could map all satisfiable formulas to "$x$" and all unsatisfiable formulas to "$x\wedge \neg x$".  (But we don't know of an efficient way to do that.)