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The two p polynomials do not coincide. I will retype the code, so that spaces are also there, this makes the code readable.

R.<x,y,z> = QQ[]
f1 = x^2 + z^2 - 1
f2 = x^2 + z^2 + (z^2 - 1)^2
J = R.ideal(f1, f2)

p = x^2 + (1/2)*y^2*z - z - 1
q = (-(1/2)*z + 1)*f1 + ((1/2)*z)*f2

print(f'Is p in J? {p in J}')
print(f'Is q in J? {q in J}')
print(f'The lift of q has the components: {q.lift(J)}')

This gives:

Is p in J? False
Is q in J? True
The lift of q has the components: [-1/2*z + 1, 1/2*z]