Constructing a PDA with an unequal number of a/b
The intuition is push the number of $a$'s then pop while reading $b$'s. The result should be non-zero: either there are still items on the stack (which we can check), or the stack has been emptied before the end of the string.

There is a problem with instructions like "$b/\varepsilon/\varepsilon$". This instruction means "read $b$ and do not push nor pop". The middle $\varepsilon$ is misleading. It does not mean that the stack is empty. It just means the PDA does not look at it.

The solution is to start by pushing a nice bottom-of-stack symbol $Z$ at the start of the computation. When we  see a $Z$ we know the bottom has been reached.