Found fic: Merlin overhears Arthur; Arthur misunderstands what Merlin overheard
Spoilers below:
Looking for a specific fic I remember reading. Set in canon. In it, Arthur has recently found out about Merlin's magic and things are still a bit rocky between them. Then Arthur goes to the tavern with Gwaine (?) and talks to him about how he is OK with Merlin's magic and also loves Merlin. Merlin overhears the first part but not the second.
The next morning, Merlin tells Arthur that he overheard their conversation and that he is very happy. Arthur thinks this means that Merlin loves him back, and so starts treating him like they are in a relationship. At some point, Merlin realises what has happened; but he is afraid to tell Arthur. He tells Arthur that he has something to tell him, and that their relationship should not go further until he does.
They both prepare gifts for each other (a magical protective item and a school, respectively; the latter involving some quick last-minute work on Gwen's part). Eventually, Merlin tells Arthur about the misunderstanding and everything is alright. Also Arthur and the knights (and eventually Merlin) carve their names onto the underside of one of the desks at the new school.
Very grateful for any help finding it! Thanks