Does that mean that it is me who has to initiate the domain transfer?

Yes, the transfer of the purchased name should be initiated by you through the registrar of your choice, using the required authorization code (“auth-code”). This code will be provided by after the purchase. Only in some cases when it is too difficult to change the registrar, may choose to push the name on the current registrar to an account under your information.

And please do not worry about the technicalities, as your registrar will surely know what to do with the auth-code.