You should check your club's/instructor's line on this.
However, I have found it depends on the airfield/airport - each has its own process and the only way to find out is to ask when you pay the landing fee on arrival.
For example, at Wycombe, we book out by completing a book in my flying club reception. They then pass the details by phone to the tower, whilst we are getting the aircraft ready. There is no further need to check if the tower received it. If they don't have the details, they ask over the radio.
At Blackbushe, I usually know how long I'm going to be staying. I pay the landing fee and book out - by phone - from the 'C' point in advance. i.e. I simply call the tower, give the registration and say I'd like to book out in advance for - say 45 minutes time.
At Bournemouth, albeit a larger airport, booking out is done by Bournemouth Handling using their computer system.
At Goodwood, booking out by anything other than requesting airfield info and taxi is unnecessary.
And so it goes on. It's always been a bit of a mystery to me why it is necessary to book out a VFR flight, but the process is a definite case of feeling your way and asking around.
Hope this helps.
HH