Continuity Error ?
After the initial battle scenes it shows (old) Bilbo beginning to write his book as Frodo walks outside to collect the post.
He brings them in to Bilbo and teases him, after discovering they are replies to his party invitations Frodo asks if he thinks Gandalf will come and Bilbo replies that he wouldn’t miss his birthday for the world.
Frodo nails the “do not disturb” sign to the gate, taking his book and explaining to Bilbo that he is heading off to the woods to greet / surprise Gandalf.
Bilbo then sits down outside and lights a pipe.
Lord of The Rings
After the initial opening scenes we see Bilbo in his house writing his book, he is disturbed by a knock at the door and he says “Frodo, door” and “Where is he ?” etc.
The scene then cuts to Frodo greeting Gandalf in the woods “A Wizard is never late” etc.
Bilbo must have forgotten moments earlier that he bid him farewell outside when he put the sign on the gate lol !