Can someone please help me understand why the second term is not added for Taylor rule?

so formula is

i∗ = r neutral + πe + 0.5 ( ‸ Y e − ‸ Y trend ) + 0.5 ( πe − π target )

but while solving π e is never included

There is some rearrangment mentioned where π e is moved to left side with target rate to get value in real terms.

does it depend on nominal or real target rate to add or not to add πe?

It depends on whether r_{neutral} is given as a real rate or a nominal rate.

Easiest way to remember is that you always want the R neutral as nominal. Therefore if they give you the real rate you need to make sure to add forecasted inflation:

Rneutral = forecast inflation + assumed equilibrium real interest rate