A reservation vecord is a book that contains all the details of reservations.
A booking = a reservation
Booked out/fully booked = the restaurant is full and can’t take any more reservations.
A special request = the caller asks for something more, e.g. a window table, ot a birthday cake etc.
Identify yourself = say who you are
A party = a group
Offered her an alternative = gave her another choice or possibility.
NB: some women prefer to use the title MS, instead of MRS or MISS