Is cost of equity the same as the required rate of return?

Mostly, yes. Cost of equity is the required rate of return on a company's stock, seen from the company's side. Investors call it the return they need to hold the shares, and the company treats it as the price of using their money. Expected return is a different idea.

Required return is the minimum an investor will accept for the risk. Expected return is what the investor actually thinks the stock will deliver, based on its price and forecasts. If CAPM says a stock needs 11% and you expect it to return 13%, the stock looks cheap to you. If you expect 9%, it looks expensive.

In equilibrium the two should match, which is why the security market line plots required return against beta. The term required rate of return also applies to bonds and projects, while cost of equity refers only to stock. A hurdle rate is related.