My understanding of the word 'contact' from an ATC perspective means that the flight has been
coordinated between the present unit, and the next, hence the next unit will have full flight details. The initial call to the next unit would be 'Odiham G-ABCD' (reg c/s phonetically, of course).
The antithesis of 'Contact' is 'Freecall'. Full flight info, in this case passed by the pilot, to the next unit. Practically speaking, the initial Freecall would wisely be transmitted 'Odiham G-ABCD'. The controller may then ask for your details by return, or after a short period of standby, when they are ready to accept your details. Even in these circumstances, a reply with the full PHACER treatment might not be welcomed, and the controller, particularly in a radar environment will ask for your flight details they
need at that tactical point in time. Eg quite often a squawk would be issued negating the need for the pilot to pass position info over the RTF. Area FISOs, eg London Information, may also selectively request info, but generally anticipate the full PHACER message.
However, controllers who are dealing with such a scenario as described so far in this thread, may use the word 'Contact' along with the phrase, 'they have your details', and this is simply to prevent the full PHACER treatment to the next unit, which may well be a very busy Area FISO, or radar controller.
Sorry this is a bit long winded, but stems from several decades of my own private flying, and working in operational ATC units at Centres and Aerodromes in the UK.
I just hope this helps!!! If not, errrm, thanks for your time!!!
In all aviation training I have received, and given, the phrase 'go ahead' must not pass your lips.

ALWAYS 'Pass your message'.
Question from me, if an individual needs RTF Phraseology coaching/brush up on the ground are there any particular rules about who can give that training? I had an AOPA ground instructor certificate, and for several years gave RTF training under the supervision of the FTO RTF Examiner. That was a few years ago now. Can anybody enlighten about any current regs about this?
Thanks.