Perfectly correct. The legal niceties have to be discussed in that, yes the Fire Certificate requires you to inform the Fire Authority concerned of your proposal to carry out any 'material alteration' to the premises concerned. The Building Control department are only required to consult the FA if they are considering 'relaxations' (in many cases they have a good relationship with the FA and would consult anyhow (workload and time permitting).
The situation you are faced with is do you rely on the fact that you put an application into the Building Control dept and they may have consulted!! A very difficult one to argue in a court, even if they did consult then the point could be raised that they informed the FA of the alterations and by default you, the occupier/owner did not and therefore acted not in compliance with the requirements, as stated in the Fire Certificate.
Long winded but my suggestion would be to inform the Fire Authority PRIOR to carrying out ANY alteration. It can be verbally, by phone, and gives you the comfort of complying with the F Cert.
PS The charges for amending a Fire Certificate following alterations is cheaper than the charge for amending following the issue of Section 8 requirements to rectify any situation.