Hermes, not Eagle, was Britain's purpose-built carrier

Did the Royal Navy have plans for purpose-built carriers before the Washington Naval Treaty, apart from Eagle?

Yes, but Eagle was itself a conversion and Hermes was the purpose-built ship; other plans existed without much urgency behind them.

Eagle began as a battleship conversion, whereas Hermes was designed as a carrier from the outset. The Royal Navy possessed other design ideas before the Washington Naval Treaty but was not pressing them rapidly into construction. Courageous and Glorious were not initially the conversions the service most wanted. Their eventual conversion reflected outside pressure and treaty-era circumstances rather than a simple absence of purpose-built thinking.