Our gaming group had an argument last week on what constituted a "sword" for magic item purposes in 5e. I argued that a glaive should be considered a sword and a polearm, as all a glaive really is is a shortsword or small scimitar blade on a stick. Another player argues that only things that "looks" like a traditional sword with an easily recognizable hilt, crossguard or handguard, blade, and pommel, should be considered a sword. Another player says that swords should be further restricted to only include slashing weapons that fit the prior description.
We all flipped through the PHB and DMG for a definition, but none of us could find anything. Is there an actual definition in the rules, or is this just another example of 5e being unfinished and requiring house rules to even be playable?