Do we need the outer discount term when implementing REINFORCE algorithm
The answer to your question is already in the text which you copied from Sutton's and Barto's book.
Sutton's and Barto's pseudocode describes the general, discounted, case. While the lecture by Silver talks about the undiscounted case. And in the undiscounted case $\gamma=1$ and therefore $\gamma^t=1$. So you can just omit it. Equation (13.8) in Sutton's and Barto's textbook also provides this simplified equation for the undiscounted case.
Also note that Monte Carlo methods are applied to episodic MDPs only. Therefore, in many cases you might not need the discounted version.