The usual course of events is as follows:
On ATC Clearance freq: From Ground, " Aircraft ABCD, Read back correct, call ground on 123.89 when fully ready"
Fully Ready means all doors and hatches closed, a tug and an engineer are connected and you are 'Fully Ready', just awaiting 'the word' from ATC.
Next call is usually on Ground Movement Control Freq:
"Ground, this is Aircraft ABCD, stand H34 fully ready"
If there are no slot times and traffic allows you will usually get the instruction to push and start, otherwise you will get a "Roger Aircraft ABCD, standby for push and start" or a repeat of your slot time so you sit and wait for them to call you back, they know who and where you are and they have to fit you in to the puzzle.
Abuse of the system may have some bearing on the use of the now common expression, "Call fully ready". I can remember sitting on a remote stand at Palma Majorca, all buttoned up and waiting to go, when the Spantax DC9 next pan called for start, he was given it, only problem was the last bus from the terminal was still to arrive, baggage still to be loaded etc. etc., we were told we were #2 to the Spantax!
There is a nightly armada of heavy jets from the Far East all heading for Europe at very adjacent departure times and the competition for levels is stiff, calling for start when doors are still open has happened, hence the request, "Call when fully ready".
Sorry, long winded but it is a very old practice in congested areas and airports.